From 5cacf4ffaf3a9621ab536195f6469f98a420f054 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 5 Sep 2026 21:55:33 +0200 Subject: Improve knowledge base --- source/know/concept/blochs-theorem/index.md | 25 +-- source/know/concept/boltzmann-equation/index.md | 20 ++- .../concept/diffie-hellman-key-exchange/index.md | 16 +- source/know/concept/dyson-equation/index.md | 38 ++-- .../concept/electric-dipole-approximation/index.md | 6 +- .../concept/equation-of-motion-theory/index.md | 7 +- source/know/concept/fabry-perot-cavity/index.md | 24 +-- .../know/concept/heaviside-step-function/index.md | 18 +- source/know/concept/heisenberg-picture/index.md | 3 +- .../know/concept/hellmann-feynman-theorem/index.md | 2 +- source/know/concept/langmuir-waves/index.md | 39 +++-- source/know/concept/legendre-transform/index.md | 16 +- source/know/concept/maxwell-relations/index.md | 14 +- .../know/concept/multi-photon-absorption/index.md | 7 +- .../concept/pauli-exclusion-principle/index.md | 119 +++++++------ source/know/concept/repetition-code/index.md | 9 +- source/know/concept/second-quantization/index.md | 192 ++++++++++++--------- source/know/concept/shors-algorithm/index.md | 62 ++++--- .../concept/sokhotski-plemelj-theorem/index.md | 4 +- .../know/concept/thermodynamic-potential/index.md | 49 +++--- source/know/concept/wkb-approximation/index.md | 140 ++++++++------- 21 files changed, 451 insertions(+), 359 deletions(-) (limited to 'source/know') diff --git a/source/know/concept/blochs-theorem/index.md b/source/know/concept/blochs-theorem/index.md index d7fcf90..c6278f3 100644 --- a/source/know/concept/blochs-theorem/index.md +++ b/source/know/concept/blochs-theorem/index.md @@ -12,14 +12,14 @@ given a potential $$V(\vb{r})$$ which is periodic on a lattice, i.e. $$V(\vb{r}) = V(\vb{r} + \vb{a})$$ for a primitive lattice vector $$\vb{a}$$, then it follows that the solutions $$\psi(\vb{r})$$ -to the time-independent Schrödinger equation -take the following form, +to the time-independent Schrödinger equation take the following form, where the function $$u(\vb{r})$$ is periodic on the same lattice, i.e. $$u(\vb{r}) = u(\vb{r} + \vb{a})$$: $$\begin{aligned} \boxed{ - \psi(\vb{r}) = u(\vb{r}) e^{i \vb{k} \cdot \vb{r}} + \psi(\vb{r}) + = u(\vb{r}) e^{i \vb{k} \cdot \vb{r}} } \end{aligned}$$ @@ -33,9 +33,11 @@ then both $$\psi(\vb{r})$$ and $$\psi(\vb{r} + \vb{a})$$ are eigenstates with the same energy: $$\begin{aligned} - \hat{H} \psi(\vb{r}) = E \psi(\vb{r}) - \qquad - \hat{H} \psi(\vb{r} + \vb{a}) = E \psi(\vb{r} + \vb{a}) + \hat{H} \psi(\vb{r}) + = E \psi(\vb{r}) + \qquad \qquad + \hat{H} \psi(\vb{r} + \vb{a}) + = E \psi(\vb{r} + \vb{a}) \end{aligned}$$ Now define the unitary translation operator $$\hat{T}(\vb{a})$$ such that @@ -52,18 +54,21 @@ $$\begin{aligned} In other words, if $$\hat{H}$$ is lattice-periodic, then it will commute with $$\hat{T}(\vb{a})$$, i.e. $$[\hat{H}, \hat{T}(\vb{a})] = 0$$. -Consequently, $$\hat{H}$$ and $$\hat{T}(\vb{a})$$ must share eigenstates $$\psi(\vb{r})$$: +Consequently, $$\hat{H}$$ and $$\hat{T}(\vb{a})$$ +must share eigenstates $$\psi(\vb{r})$$: $$\begin{aligned} - \hat{H} \:\psi(\vb{r}) = E \:\psi(\vb{r}) + \hat{H} \psi(\vb{r}) + = E \psi(\vb{r}) \qquad \qquad - \hat{T}(\vb{a}) \:\psi(\vb{r}) = \tau \:\psi(\vb{r}) + \hat{T}(\vb{a}) \psi(\vb{r}) + = \tau \psi(\vb{r}) \end{aligned}$$ Since $$\hat{T}$$ is unitary, its eigenvalues $$\tau$$ must have the form $$e^{i \theta}$$, with $$\theta$$ real. Therefore a translation by $$\vb{a}$$ causes a phase shift, -for some vector $$\vb{k}$$: +so there exists a vector $$\vb{k}$$ such that: $$\begin{aligned} \psi(\vb{r} + \vb{a}) diff --git a/source/know/concept/boltzmann-equation/index.md b/source/know/concept/boltzmann-equation/index.md index 5f4add0..3821512 100644 --- a/source/know/concept/boltzmann-equation/index.md +++ b/source/know/concept/boltzmann-equation/index.md @@ -67,7 +67,7 @@ but unfortunately also quite difficult to work with. In addition, $$f$$ is a 7-dimensional function, so the BTE is already hard to solve without collisions! We only present the simplest case, -known as the **Bhatnagar-Gross-Krook approximation**: +the **Bhatnagar-Gross-Krook approximation**: if the equilibrium state $$f_0(\vb{r}, \vb{v})$$ is known, then each collision brings the system closer to $$f_0$$: @@ -90,14 +90,15 @@ $$\begin{aligned} n(\vb{r}, t) = \int_{-\infty}^\infty f(\vb{r}, \vb{v}, t) \dd{\vb{v}} \end{aligned}$$ -Consequently, a purely velocity-dependent quantity $$Q(\vb{v})$$ can be averaged like so: +Consequently, a purely velocity-dependent quantity $$Q(\vb{v})$$ +can be averaged like so: $$\begin{aligned} - \Expval{Q} - = \frac{1}{n} \int_{-\infty}^\infty Q(\vb{r}, \vb{v}, t) \: f(\vb{r}, \vb{v}, t) \dd{\vb{v}} + \Expval{Q}\!(\vb{r}, t) + = \frac{1}{n} \int_{-\infty}^\infty Q(\vb{v}) \: f(\vb{r}, \vb{v}, t) \dd{\vb{v}} \end{aligned}$$ -With that in mind, we multiply the collisionless BTE equation by $$Q(\vb{v})$$ and integrate, +With that in mind, we multiply the collisionless BTE by $$Q(\vb{v})$$ and integrate, assuming that $$\vb{F}$$ does not depend on $$\vb{v}$$: $$\begin{aligned} @@ -136,7 +137,8 @@ $$\begin{aligned} If we set $$Q = m$$, then the mass density $$\rho = n \Expval{Q}$$, and we find that the **zeroth moment** of the BTE describes conservation of mass, -where $$\vb{V} \equiv \Expval{\vb{v}} = \int \vb{v} f \dd{\vb{v}}$$ is the fluid velocity: +where $$\vb{V} \equiv \Expval{\vb{v}} = n^{-1} \int \vb{v} f \dd{\vb{v}}$$ +is the fluid velocity: $$\begin{aligned} \boxed{ @@ -231,7 +233,8 @@ $$\begin{aligned} {% include proof/start.html id="proof-moment2" -%} -We insert $$Q = m |\vb{v}|^2 / 2$$ into our prototype and recognize $$\rho$$ wherever possible: +We insert $$Q = m |\vb{v}|^2 / 2$$ into our prototype +and recognize $$\rho$$ wherever possible: $$\begin{aligned} 0 @@ -244,7 +247,8 @@ $$\begin{aligned} - \frac{\vb{F}}{2} \cdot \bigg( n \Expval{\pdv{|\vb{v}|^2}{\vb{v}}} \bigg) \end{aligned}$$ -We handle these terms one by one. Substituting $$\vb{v} = \vb{V} + \vb{w}$$ in the first gives: +We handle these terms one by one. +Substituting $$\vb{v} = \vb{V} + \vb{w}$$ in the first gives: $$\begin{aligned} \Expval{|\vb{v}|^2} diff --git a/source/know/concept/diffie-hellman-key-exchange/index.md b/source/know/concept/diffie-hellman-key-exchange/index.md index 3525881..a6e0894 100644 --- a/source/know/concept/diffie-hellman-key-exchange/index.md +++ b/source/know/concept/diffie-hellman-key-exchange/index.md @@ -32,19 +32,22 @@ there is no efficient algorithm to recover $$n$$. Suppose that Alice and Bob want to exchange encrypted data in the future, so they need to agree on an encryption key to use. -However, they can only exchange messages with each other over +However, they can only exchange messages over an insecure channel, which is being eavesdropped. After they publicly agree on the values of $$g$$ and $$p$$, -Alice and Bob each choose a secret number from $$\{0, ..., p \!-\! 2\}$$, respectively $$a$$ and $$b$$, +Alice and Bob each choose a secret number from $$\{0, ..., p \!-\! 2\}$$, +respectively $$a$$ and $$b$$, and then privately calculate $$A$$ and $$B$$ as follows: $$\begin{aligned} A - \equiv g^a \bmod p - \qquad \qquad + &\equiv f(a) + = g^a \bmod p + \\ B - \equiv g^b \bmod p + &\equiv f(b) + = g^b \bmod p \end{aligned}$$ Finally, they transmit these numbers $$A$$ and $$B$$ @@ -69,8 +72,7 @@ but cannot recover $$a$$ or $$b$$. This assumption is just that: an assumption. So far, nobody has been able to prove or disprove it for classical computation. -However, for quantum computers, -it has already been *dis*proven! +However, for quantum computers, it has already been *dis*proven! In this case, another method must be used, for example the [BB84 protocol](/know/concept/bb84-protocol/). diff --git a/source/know/concept/dyson-equation/index.md b/source/know/concept/dyson-equation/index.md index ae9eb35..03be06f 100644 --- a/source/know/concept/dyson-equation/index.md +++ b/source/know/concept/dyson-equation/index.md @@ -25,8 +25,8 @@ $$\begin{aligned} = \delta(\vb{r} - \vb{r}') \: \delta(t - t') \end{aligned}$$ -From this, we define the inverse $$\hat{G}{}_0^{-1}(\vb{r}, t)$$ -as follows, so that $$\hat{G}{}_0^{-1} G_0 = \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t')$$: +From this, we define the inverse $$\hat{G}{}_0^{-1}(\vb{r}, t)$$ as follows, +so that $$\hat{G}{}_0^{-1} G_0 = \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t')$$: $$\begin{aligned} \hat{G}{}_0^{-1}(\vb{r}, t) @@ -35,16 +35,15 @@ $$\begin{aligned} Note that $$\hat{G}{}_0^{-1}$$ is an operator, while $$G_0$$ is a function. For the sake of consistency, we thus define -the operator $$\hat{G}_0(\vb{r}, t)$$ -as a multiplication by $$G_0$$ -and integration over $$\vb{r}'$$ and $$t'$$: +its operator version $$\hat{G}_0(\vb{r}, t)$$ +as a multiplication by $$G_0$$ and integration over $$\vb{r}'$$ and $$t'$$: $$\begin{aligned} \hat{G}_0(\vb{r}, t) \: f - \equiv \iint_{-\infty}^\infty G_0(\vb{r}, t; \vb{r}', t') \: f(\vb{r}', t') \: \dd{\vb{r}}' \dd{t'} + \equiv \iint_{-\infty}^\infty G_0(\vb{r}, t; \vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}}' \dd{t'} \end{aligned}$$ -For an arbitrary function $$f(\vb{r}, t)$$, +Where $$f(\vb{r}, t)$$ is an arbitrary function, so that $$\hat{G}{}_0^{-1} \hat{G}_0 = \hat{G}_0 \hat{G}{}_0^{-1} = 1$$. Moving on, the Schrödinger equation can be rewritten like so, using $$\hat{G}{}_0^{-1}$$: @@ -61,7 +60,7 @@ by solving the defining equation above. Suppose we now add a more complicated and possibly time-dependent term $$\hat{H}_1(\vb{r}, t)$$, in which case the corresponding fundamental solution -$$G(\vb{r}, \vb{r}', t, t')$$ satisfies: +$$G(\vb{r}, \vb{r}', t, t')$$ (note the lack of a $$0$$ subscript) satisfies: $$\begin{aligned} \delta(\vb{r} - \vb{r}') \: \delta(t - t') @@ -72,7 +71,7 @@ $$\begin{aligned} This equation is typically too complicated to solve, so we would like an easier way to calculate this new $$G$$. -The perturbed wavefunction $$\Psi(\vb{r}, t)$$ +Clearly, the perturbed wavefunction $$\Psi(\vb{r}, t)$$ satisfies the Schrödinger equation: $$\begin{aligned} @@ -80,9 +79,8 @@ $$\begin{aligned} = 0 \end{aligned}$$ -We know that $$\hat{G}{}_0^{-1} \Psi_0 = 0$$, -which we put on the right, -and then we apply $$\hat{G}_0$$ in front: +We know that $$\hat{G}{}_0^{-1} \Psi_0 = 0$$ from earlier, +which we put on the right, and then apply $$\hat{G}_0$$ to it: $$\begin{aligned} \hat{G}_0^{-1} \Psi - \hat{H}_1 \Psi @@ -110,7 +108,8 @@ $$\begin{aligned} \end{aligned}$$ The parenthesized expression clearly has the same recursive pattern, -so we denote it by $$\hat{G}$$ and write the so-called **Dyson equation**: +so we denote it by $$\hat{G}$$ (an operator, not the function $$G$$) +and write the so-called **Dyson equation**: $$\begin{aligned} \boxed{ @@ -133,8 +132,8 @@ $$\begin{aligned} This relation is equivalent to the Schrödinger equation. So now we have the operator $$\hat{G}(\vb{r}, t)$$, but what about the fundamental solution function $$G(\vb{r}, t; \vb{r}', t')$$? -Let us take its definition, multiply it by an arbitrary $$f(\vb{r}, t)$$, -and integrate over $$G$$'s second argument pair: +Let us take the latter's definition and multiply it by an arbitrary $$f(\vb{r}, t)$$, +and then integrate over $$G$$'s second argument pair: $$\begin{aligned} \iint \big( \hat{G}{}_0^{-1} \!-\! \hat{H}_1 \big) G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'} @@ -143,8 +142,7 @@ $$\begin{aligned} \end{aligned}$$ Where we have hidden the arguments $$(\vb{r}, t)$$ for brevity. -We now apply $$\hat{G}_0(\vb{r}, t)$$ to this equation -(which contains an integral over $$t''$$ independent of $$t'$$): +We apply $$\hat{G}_0(\vb{r}, t)$$ to this equation: $$\begin{aligned} \hat{G}_0 f @@ -154,8 +152,10 @@ $$\begin{aligned} \end{aligned}$$ Here, the shape of Dyson's equation is clearly recognizable, -so we conclude that, as expected, the operator $$\hat{G}$$ -is defined as multiplication by the function $$G$$ followed by integration: +so we conclude that the operator $$\hat{G}$$ +is defined as multiplication by the function $$G$$ followed by integration, +exactly analogously to $$\hat{G}_0$$ and $$G_0$$, +which should not be a big surprise: $$\begin{aligned} \hat{G}(\vb{r}, t) \: f(\vb{r}, t) diff --git a/source/know/concept/electric-dipole-approximation/index.md b/source/know/concept/electric-dipole-approximation/index.md index 06f0f45..393b875 100644 --- a/source/know/concept/electric-dipole-approximation/index.md +++ b/source/know/concept/electric-dipole-approximation/index.md @@ -138,10 +138,10 @@ $$\begin{aligned} \\ &= - (- i i) q \omega_0 \vu{x} \cdot \vb{A}_0 \exp(- i \omega t) \\ - &\approx - \vu{d} \cdot \vb{E}_0 \exp(- i \omega t) + &\approx - \vu{p} \cdot \vb{E}_0 \exp(- i \omega t) \end{aligned}$$ -Where $$\vu{d} \equiv q \vu{x}$$ is +Where $$\vu{p} \equiv q \vu{x}$$ is the **transition dipole moment operator** of the electron, hence the name *electric dipole approximation*. Finally, we take the real part, yielding: @@ -150,7 +150,7 @@ $$\begin{aligned} \boxed{ \begin{aligned} \hat{H}_1(t) - &= - \vu{d} \cdot \vb{E}(t) + &= - \vu{p} \cdot \vb{E}(t) \\ &= - q \vu{x} \cdot \vb{E}_0 \cos(\omega t) \end{aligned} diff --git a/source/know/concept/equation-of-motion-theory/index.md b/source/know/concept/equation-of-motion-theory/index.md index c1ed8da..76cee81 100644 --- a/source/know/concept/equation-of-motion-theory/index.md +++ b/source/know/concept/equation-of-motion-theory/index.md @@ -100,7 +100,8 @@ $$\begin{aligned} \\ &= \sum_{\nu' \nu''} u_{\nu' \nu''} \Big( \delta_{\nu \nu'} \hat{f}_{\!\nu''} - 2 \acomm{\hat{f}_{\!\nu'}^\dagger}{\hat{f}_{\!\nu}} \hat{f}_{\!\nu''} \Big) - = - \sum_{\nu''} u_{\nu \nu''} \hat{f}_{\!\nu''} + \\ + &= - \sum_{\nu''} u_{\nu \nu''} \hat{f}_{\!\nu''} \end{aligned}$$ {% include proof/end.html id="proof-commutator" %} @@ -160,7 +161,7 @@ $$\begin{aligned} \end{aligned}$$ We take the [Fourier transform](/know/concept/fourier-transform/) -$$(t \!-\! t') \to (\omega + i \eta)$$, where $$\eta \to 0^+$$ ensures convergence: +$$(t - t') \to (\omega + i \eta)$$, where $$\eta \to 0^+$$ ensures convergence: $$\begin{aligned} \sum_{\nu''} \Big( \hbar \delta_{\nu \nu''} (\omega + i \eta) - u_{\nu \nu''} \Big) G^R_{\nu'' \nu'}(\omega) @@ -178,7 +179,7 @@ $$\begin{aligned} \end{aligned}$$ For a non-interacting, time-independent Hamiltonian, -we therefore arrive at: +we thus arrive at the famous result: $$\begin{aligned} \boxed{ diff --git a/source/know/concept/fabry-perot-cavity/index.md b/source/know/concept/fabry-perot-cavity/index.md index d5ea0ea..c648549 100644 --- a/source/know/concept/fabry-perot-cavity/index.md +++ b/source/know/concept/fabry-perot-cavity/index.md @@ -10,11 +10,13 @@ layout: "concept" --- In its simplest form, a **Fabry-Pérot cavity** -is a region of light-transmitting medium surrounded by two mirrors, -which may transmit some of the incoming light. -Such a setup can be used as e.g. an interferometer or a laser cavity. +is a region of light-transmitting medium surrounded by two parallel mirrors, +which may let some of the light escape. +Such a setup can be used as e.g. a laser cavity or an interferometer. +Below, we treat this simple system as an exercise +for calculating *quasinormal modes* in 1D, +i.e. modes with complex resonances. -Below, we calculate its quasinormal modes in 1D. We divide the $$x$$-axis into three domains: left $$L$$, center $$C$$, and right $$R$$. The cavity $$C$$ has length $$\ell$$ and is centered on $$x = 0$$. Let $$n_L$$, $$n_C$$ and $$n_R$$ be the respective domains' refractive indices: @@ -95,8 +97,8 @@ $$\begin{aligned} \end{bmatrix} \end{aligned}$$ -We do not want to simply satisfy this equation -by setting $$A_1$$, $$A_2$$, $$A_3$$ and $$A_4$$, +We do not want to satisfy this equation +by simply setting $$A_1$$, $$A_2$$, $$A_3$$ and $$A_4$$, so we demand that the system matrix is not invertible, i.e. its determinant is zero: @@ -116,7 +118,9 @@ $$\begin{aligned} - 2 n_C (n_L + n_R) \cos(k_m n_C \ell) \end{aligned}$$ -Finally, some further rearranging gives a convenient transcendental equation: +Finally, some further rearranging gives a convenient transcendental equation, +keeping in mind that $$k_m$$ and the indices $$n_L$$, $$n_C$$ and $$n_R$$ +are generally complex numbers: $$\begin{aligned} \boxed{ @@ -223,9 +227,9 @@ $$\begin{aligned} &= (1 - r_R) A_3 e^{i k_m (n_C - n_R) \ell/2} \end{aligned}$$ -Note that we have not demanded continuity of the electric field. -This is because the mirrors are infinitely thin "magic" planes; -had we instead included the full microscopic mirror structure, +Note that we have not demanded continuity of the electric field, +because the mirrors are infinitely thin "magic" planes in this case. +If we had instead included the full microscopic mirror structure, then we would have demanded continuity as before. diff --git a/source/know/concept/heaviside-step-function/index.md b/source/know/concept/heaviside-step-function/index.md index 9f5d4ec..6412914 100644 --- a/source/know/concept/heaviside-step-function/index.md +++ b/source/know/concept/heaviside-step-function/index.md @@ -45,15 +45,15 @@ $$\begin{aligned} \end{aligned}$$ The [Fourier transform](/know/concept/fourier-transform/) -of $$\Theta(t)$$ is as follows, -where $$\pv{}$$ is the Cauchy principal value, +of $$\Theta(t)$$ is as follows, where $$\mathcal{P}$$ +is the [Cauchy principal value](/know/concept/cauchy-principal-value/), $$A$$ and $$s$$ are constants from the FT's definition, and $$\mathrm{sgn}$$ is the signum function: $$\begin{aligned} \boxed{ \tilde{\Theta}(\omega) - = \frac{A}{|s|} \Big( \pi \delta(\omega) + i \: \mathrm{sgn}(s) \pv{\frac{1}{\omega}} \Big) + = \frac{A}{|s|} \Big( \pi \delta(\omega) + i \:\mathrm{sgn}(s) \:\mathcal{P} \frac{1}{\omega} \Big) } \end{aligned}$$ @@ -77,18 +77,18 @@ $$\begin{aligned} \end{aligned}$$ The first term is proportional to the Dirac delta function. -The second integral is problematic, so we take the Cauchy principal value $$\pv{}$$ -and look up the integral: +The second integral is problematic, so we take +the Cauchy principal value $$\mathcal{P}$$ and look up the integral: $$\begin{aligned} \tilde{\Theta}(\omega) - &= A \pi \delta(s \omega) + \frac{A}{2} \pv{\int_{-\infty}^\infty \mathrm{sgn}(t) \exp(i s \omega t) \dd{t}} - = \frac{A}{|s|} \pi \delta(\omega) + i \frac{A}{s} \pv{\frac{1}{\omega}} + &= A \pi \delta(s \omega) + \frac{A}{2} \:\mathcal{P}\! \int_{-\infty}^\infty \mathrm{sgn}(t) \exp(i s \omega t) \dd{t} + = \frac{A}{|s|} \pi \delta(\omega) + i \frac{A}{s} \:\mathcal{P} \frac{1}{\omega} \end{aligned}$$ {% include proof/end.html id="proof-fourier" %} -The use of $$\pv{}$$ without an integral is an abuse of notation, +The use of $$\mathcal{P}$$ without an integral is an abuse of notation, and means that this result only makes sense when wrapped in an integral. -Formally, $$\pv{\{1 / \omega\}}$$ is a [Schwartz distribution](/know/concept/schwartz-distribution/). +Formally, $$\mathcal{P}\{1 / \omega\}$$ is a [Schwartz distribution](/know/concept/schwartz-distribution/). diff --git a/source/know/concept/heisenberg-picture/index.md b/source/know/concept/heisenberg-picture/index.md index 54bf397..3ffe29a 100644 --- a/source/know/concept/heisenberg-picture/index.md +++ b/source/know/concept/heisenberg-picture/index.md @@ -118,7 +118,7 @@ This result is arguably more intuitive than the Schrödinger picture, because it allows us to think about observables (i.e. operators) in a more classical way. For example, inserting the position $$\hat{X}$$ and momentum $$\hat{P} = - i \hbar \: \idv{}{\hat{X}}$$ -gives the following Newton-style relations: +gives the following Newton-style relations (details omitted): $$\begin{aligned} \dv{\hat{X}}{t} @@ -130,7 +130,6 @@ $$\begin{aligned} = - \dv{V(\hat{X})}{\hat{X}} \end{aligned}$$ -Where the commutators have been treated as known. These equations would not be valid in the Schrödinger picture, unless we took their expectation value to get [Ehrenfest's theorem](/know/concept/ehrenfests-theorem/). diff --git a/source/know/concept/hellmann-feynman-theorem/index.md b/source/know/concept/hellmann-feynman-theorem/index.md index c6bf720..d02b285 100644 --- a/source/know/concept/hellmann-feynman-theorem/index.md +++ b/source/know/concept/hellmann-feynman-theorem/index.md @@ -72,7 +72,7 @@ $$\begin{aligned} } \end{aligned}$$ -While for $$m \neq n$$, we get the **Epstein generalization** +While for $$m \neq n$$, we get the (unfortunately named) **Epstein generalization** of the Hellmann-Feynman theorem, which is for example relevant for the [Berry phase](/know/concept/berry-phase/): diff --git a/source/know/concept/langmuir-waves/index.md b/source/know/concept/langmuir-waves/index.md index 2dbce8f..736ef71 100644 --- a/source/know/concept/langmuir-waves/index.md +++ b/source/know/concept/langmuir-waves/index.md @@ -54,13 +54,13 @@ are assumed to satisfy: $$\begin{aligned} \pdv{n_{e0}}{t} = 0 - \qquad + \qquad \quad \pdv{\vb{u}_{e0}}{t} = 0 - \qquad + \qquad \quad \nabla n_{e0} = 0 - \qquad + \qquad \quad \vb{u}_{e0} = 0 - \qquad + \qquad \quad \vb{E}_0 = 0 \end{aligned}$$ @@ -73,8 +73,7 @@ $$\begin{aligned} \\ &= \pdv{n_{e1}}{t} + \nabla \cdot \Big( n_{e0} \vb{u}_{e1} + n_{e1} \vb{u}_{e1} \Big) \\ - &\approx \pdv{n_{e1}}{t} + \nabla \cdot (n_{e0} \vb{u}_{e1}) - = \pdv{n_{e1}}{t} + n_{e0} \nabla \cdot \vb{u}_{e1} + &\approx \pdv{n_{e1}}{t} + n_{e0} \nabla \cdot \vb{u}_{e1} \end{aligned}$$ Likewise, we insert it into Gauss' law, @@ -83,7 +82,7 @@ and use the plasma's quasi-neutrality $$n_i = n_{e0}$$ to get: $$\begin{aligned} \varepsilon_0 \nabla \cdot \big( \vb{E}_0 \!+\! \vb{E}_1 \big) = q_e (n_{e0} + n_{e1} - n_i) - \quad \implies \quad + \qquad \implies \qquad \varepsilon_0 \nabla \cdot \vb{E}_1 = q_e n_{e1} \end{aligned}$$ @@ -106,12 +105,13 @@ Inserting this into the continuity equation and Gauss' law yields, respectively: $$\begin{aligned} - i \omega n_{e1} = - i n_{e0} \vb{k} \cdot \vb{u}_{e1} - \qquad \quad + \qquad \qquad -\! i \varepsilon_0 \vb{k} \cdot \vb{E}_1 = q_e n_{e1} \end{aligned}$$ +These form a system of equations to be solved. However, there are three unknowns $$n_{e1}$$, $$\vb{u}_{e1}$$ and $$\vb{E}_1$$, -so one more equation is needed. +so one more equation is needed before we can do so. @@ -180,7 +180,8 @@ the oscillation is stationary. ## Warm Langmuir waves Next, we generalize this result to nonzero $$T_e$$, -in which case the pressure $$p_e$$ is involved: +in which case the pressure $$p_e$$ is involved, +so the electron momentum equation is given by: $$\begin{aligned} m_e n_{e0} \pdv{\vb{u}_{e1}}{t} @@ -198,10 +199,11 @@ $$\begin{aligned} \end{aligned}$$ With this, insertion of our plane-wave ansatz -into the electron equation results in: +into the momentum equation results in: $$\begin{aligned} - -i \omega m_e n_{e0} \vb{u}_{e1} = q_e n_{e0} \vb{E}_1 - i \gamma k_B T_e n_{e1} \vb{k} + -i \omega m_e n_{e0} \vb{u}_{e1} + = q_e n_{e0} \vb{E}_1 - i \gamma k_B T_e n_{e1} \vb{k} \end{aligned}$$ Which once again closes the system of three equations. @@ -209,7 +211,8 @@ Solving for $$\omega^2$$ then gives: $$\begin{aligned} \omega^2 - = \frac{\omega n_{e0}}{n_{e1}} \vb{k} \cdot \vb{u}_{e1} + &= \frac{\omega n_{e0}}{n_{e1}} \vb{k} \cdot \vb{u}_{e1} + \\ &= \frac{i \omega n_{e0}}{\omega n_{e0} m_e n_{e1}} \vb{k} \cdot \Big( q_e n_{e0} \vb{E}_1 - i \gamma k_B T_e n_{e1} \vb{k} \Big) \\ &= \frac{n_{e0} q_e^2}{\varepsilon_0 m_e} - \frac{i \omega}{\omega m_e n_{e1}} i \gamma k_B T_e n_{e1} \big(\vb{k} \cdot \vb{k}\big) @@ -235,13 +238,13 @@ $$\begin{aligned} \end{aligned}$$ Unlike for $$T_e = 0$$, these "warm" waves do propagate, -carrying information at group velocity $$v_g$$, -which, in the limit of large $$k$$, is given by: +because $$k$$ appears in the dispersion relation. +They carry information at group velocity $$v_g = \ipdv{w}{k}$$, +which in the limit of large $$k$$ becomes: $$\begin{aligned} - v_g - = \pdv{\omega}{k} - \to \sqrt{\frac{3 k_B T_e}{m_e}} + \lim_{k \to \infty} v_g + = \sqrt{\frac{3 k_B T_e}{m_e}} \end{aligned}$$ This is the root-mean-square velocity of the diff --git a/source/know/concept/legendre-transform/index.md b/source/know/concept/legendre-transform/index.md index d09613f..0d168aa 100644 --- a/source/know/concept/legendre-transform/index.md +++ b/source/know/concept/legendre-transform/index.md @@ -11,9 +11,8 @@ layout: "concept" The **Legendre transform** of a function $$f(x)$$ is a new function $$L(f')$$, which depends only on the derivative $$f'(x)$$ of $$f(x)$$, and from which the original $$f(x)$$ can be reconstructed. -The point is that $$L(f')$$ contains the same information as $$f(x)$$, -just in a different form, -analogously to e.g. the [Fourier transform](/know/concept/fourier-transform/). +The point is that $$L(f')$$ contains the same information as $$f(x)$$ +in a different form, like e.g. a [Fourier transform](/know/concept/fourier-transform/). Let us choose an arbitrary point $$x_0 \in [a, b]$$ in the domain of $$f(x)$$. Consider a line $$y(x)$$ tangent to $$f(x)$$ at $$x = x_0$$, @@ -23,18 +22,17 @@ $$\begin{aligned} y(x) &= f'(x_0) (x - x_0) + f(x_0) \\ - &= f'(x_0) \: x - C + &= f'(x_0) \: x - C(x_0) \end{aligned}$$ -Where $$C \equiv f'(x_0) \: x_0 - f(x_0)$$. +Where $$C(x) \equiv f'(x) \: x - f(x)$$. We now define the *Legendre transform* $$L(f')$$, -such that for all $$x_0 \in [a, b]$$ we have $$L(f'(x_0)) = C$$ -(some authors use $$-C$$ instead). -Renaming $$x_0$$ to $$x$$: +such that for all $$x_0 \in [a, b]$$ we have $$L(f'(x_0)) = C(x_0)$$ +(some authors use $$-C$$ instead): $$\begin{aligned} L(f'(x)) - &= f'(x) \: x - f(x) + &\equiv f'(x) \: x - f(x) \end{aligned}$$ We want this function to depend only on the derivative $$f'$$, diff --git a/source/know/concept/maxwell-relations/index.md b/source/know/concept/maxwell-relations/index.md index 892ced1..f51acea 100644 --- a/source/know/concept/maxwell-relations/index.md +++ b/source/know/concept/maxwell-relations/index.md @@ -9,7 +9,7 @@ layout: "concept" --- The **Maxwell relations** are a useful set of relations in thermodynamics. -They arise from the fact that the order of differentiation is irrelevant +They arise from the fact that the ordering of differentiation is irrelevant for well-behaved functions (sometimes known as the *Schwarz theorem*), applied to the [thermodynamic potentials](/know/concept/thermodynamic-potential/). @@ -54,7 +54,7 @@ $$\begin{aligned} = \Big( \pdv{B}{x} \Big)_y^{-1} \end{aligned}$$ -The following quantities are useful to rewrite some of the Maxwell relations: +The following quantities can be useful to rewrite some of the Maxwell relations: the iso-$$P$$ thermal expansion coefficient $$\alpha$$, the iso-$$T$$ combressibility $$\kappa_T$$, the iso-$$S$$ combressibility $$\kappa_S$$, @@ -73,6 +73,10 @@ $$\begin{gathered} C_P \equiv T \Big( \pdv{S}{T} \Big)_{P,N} \end{gathered}$$ +But for simplicity and brevity, +we will not do any such rewriting in this article. + + ## Internal energy @@ -116,6 +120,7 @@ $$\begin{gathered} \end{gathered}$$ + ## Enthalpy The following Maxwell relations can be derived @@ -158,6 +163,7 @@ $$\begin{gathered} \end{gathered}$$ + ## Helmholtz free energy The following Maxwell relations can be derived @@ -200,6 +206,7 @@ $$\begin{gathered} \end{gathered}$$ + ## Gibbs free energy The following Maxwell relations can be derived @@ -242,10 +249,11 @@ $$\begin{gathered} \end{gathered}$$ + ## Landau potential The following Maxwell relations can be derived -from the Gibbs free energy $$\Omega(T, V, \mu)$$: +from the Landau potential $$\Omega(T, V, \mu)$$: $$\begin{gathered} - \mpdv{\Omega}{V}{T} = diff --git a/source/know/concept/multi-photon-absorption/index.md b/source/know/concept/multi-photon-absorption/index.md index 80dbc9b..481c19d 100644 --- a/source/know/concept/multi-photon-absorption/index.md +++ b/source/know/concept/multi-photon-absorption/index.md @@ -30,7 +30,6 @@ Here, we have made the to neglect the $$e^{i \omega t}$$ term, because it turns out to be irrelevant in this discussion. - We call the ground state $$\Ket{0}$$, but other than that, the other states need *not* be sorted by energy. However, we demand that the following holds @@ -187,7 +186,7 @@ i.e. for any odd-numbered final state $$\Ket{u}$$. ## Two-photon absorption Next, we go to second-order perturbation theory. -Based on the previous result, this time +Thanks to the previous result $$c_e^{(1)}(t) = 0$$, this time all odd-numbered states $$\Ket{u}$$ are unaffected: $$\begin{aligned} @@ -248,7 +247,7 @@ two identical photons $$\hbar \omega$$ are absorbed simultaneously to bridge the energy gap $$\hbar \omega_{e0}$$. Surprisingly, such a transition can only occur when $$\matrixel{e}{\vu{p}}{0} = 0$$, i.e. for any even-numbered final state $$\Ket{e}$$. -Notice that the rate is proportional to $$|\vb{E}|^4$$, +The rate is proportional to $$|\vb{E}|^4$$, so this effect is only noticeable at high light intensities. @@ -339,7 +338,7 @@ due to the dependence on $$\vb{E}$$. If $$N$$ is odd, only odd-numbered destinations $$\Ket{u}$$ are allowed (assuming the electron starts in the ground state $$\Ket{0}$$), and if $$N$$ is even, only even-numbered destinations $$\Ket{e}$$. -Note that nothing has been said about the energies of these states +Nothing has been said about the energies of these states (other than $$\Ket{0}$$ being the minimum); everything is determined by the matrix elements $$\matrixel{f}{\vu{p}}{i}$$. diff --git a/source/know/concept/pauli-exclusion-principle/index.md b/source/know/concept/pauli-exclusion-principle/index.md index 5b83b69..15130d9 100644 --- a/source/know/concept/pauli-exclusion-principle/index.md +++ b/source/know/concept/pauli-exclusion-principle/index.md @@ -8,57 +8,69 @@ categories: layout: "concept" --- -In quantum mechanics, the **Pauli exclusion principle** is a theorem with -profound consequences for how the world works. +In quantum mechanics, the **Pauli exclusion principle** is a theorem +with profound consequences for how the world works. Suppose we have a composite state -$$\ket{x_1}\ket{x_2} = \ket{x_1} \otimes \ket{x_2}$$, where the two -identical particles $$x_1$$ and $$x_2$$ each can occupy the same two allowed -states $$a$$ and $$b$$. We then define the permutation operator $$\hat{P}$$ as -follows: +$$\ket{x_1}\ket{x_2} = \ket{x_1} \otimes \ket{x_2}$$, +where the two identical particles $$x_1$$ and $$x_2$$ +each can occupy the same two allowed states $$a$$ and $$b$$. +We then define the permutation operator $$\hat{P}$$ as follows: $$\begin{aligned} - \hat{P} \Ket{a}\Ket{b} = \Ket{b}\Ket{a} + \hat{P} \Ket{a}\Ket{b} + = \Ket{b}\Ket{a} \end{aligned}$$ -That is, it swaps the states of the particles. Obviously, swapping the -states twice simply gives the original configuration again, so: +That is, it swaps the states of the particles. +Obviously, swapping the states twice simply gives the original configuration again, so: $$\begin{aligned} - \hat{P}^2 \Ket{a}\Ket{b} = \Ket{a}\Ket{b} + \hat{P}^2 \Ket{a}\Ket{b} + = \Ket{a}\Ket{b} \end{aligned}$$ -Therefore, $$\Ket{a}\Ket{b}$$ is an eigenvector of $$\hat{P}^2$$ with -eigenvalue $$1$$. Since $$[\hat{P}, \hat{P}^2] = 0$$, $$\Ket{a}\Ket{b}$$ -must also be an eigenket of $$\hat{P}$$ with eigenvalue $$\lambda$$, -satisfying $$\lambda^2 = 1$$, so we know that $$\lambda = 1$$ or $$\lambda = -1$$: +Therefore, $$\Ket{a}\Ket{b}$$ is an eigenvector of $$\hat{P}^2$$ with eigenvalue $$1$$. +Since $$[\hat{P}, \hat{P}^2] = 0$$, +$$\Ket{a}\Ket{b}$$ must also be an eigenket of $$\hat{P}$$ +with eigenvalue $$\lambda$$, satisfying $$\lambda^2 = 1$$, +so we know that $$\lambda = 1$$ or $$\lambda = -1$$: $$\begin{aligned} - \hat{P} \Ket{a}\Ket{b} = \lambda \Ket{a}\Ket{b} + \hat{P} \Ket{a}\Ket{b} + = \lambda \Ket{a}\Ket{b} \end{aligned}$$ -As it turns out, in nature, each class of particle has a single -associated permutation eigenvalue $$\lambda$$, or in other words: whether -$$\lambda$$ is $$-1$$ or $$1$$ depends on the type of particle that $$x_1$$ -and $$x_2$$ are. Particles with $$\lambda = -1$$ are called -**fermions**, and those with $$\lambda = 1$$ are known as **bosons**. We -define $$\hat{P}_f$$ with $$\lambda = -1$$ and $$\hat{P}_b$$ with -$$\lambda = 1$$, such that: +As it turns out, in nature, each type of particle has a single +associated permutation eigenvalue $$\lambda$$, or in other words: +whether $$\lambda$$ is $$-1$$ or $$1$$ depends on +the type of particle that $$x_1$$ and $$x_2$$ are. +Particles with $$\lambda = -1$$ are called **fermions**, +and those with $$\lambda = 1$$ are known as **bosons**. +We define $$\hat{P}_f$$ with $$\lambda = -1$$ and $$\hat{P}_b$$ with $$\lambda = 1$$, +such that: $$\begin{aligned} - \hat{P}_f \Ket{a}\Ket{b} = \Ket{b}\Ket{a} = - \Ket{a}\Ket{b} - \qquad - \hat{P}_b \Ket{a}\Ket{b} = \Ket{b}\Ket{a} = \Ket{a}\Ket{b} + \hat{P}_f \Ket{a}\Ket{b} + &= \Ket{b}\Ket{a} + = - \Ket{a}\Ket{b} + \\ + \hat{P}_b \Ket{a}\Ket{b} + &= \Ket{b}\Ket{a} + = \Ket{a}\Ket{b} \end{aligned}$$ -Another fundamental fact of nature is that identical particles cannot be -distinguished by any observation. Therefore it is impossible to tell -apart $$\Ket{a}\Ket{b}$$ and the permuted state $$\Ket{b}\Ket{a}$$, -regardless of the eigenvalue $$\lambda$$. There is no physical difference! - -But this does not mean that $$\hat{P}$$ is useless: despite not having any -observable effect, the resulting difference between fermions and bosons -is absolutely fundamental. Consider the following superposition state, +Another fundamental fact is that identical particles +cannot be distinguished by any observation. +Therefore, it is impossible to tell apart $$\Ket{a}\Ket{b}$$ +and the permuted state $$\Ket{b}\Ket{a}$$, +regardless of the eigenvalue $$\lambda$$. +There is no physical difference! + +But this does not mean that $$\hat{P}$$ is useless: +despite not having any observable effect, +the resulting difference between fermions and bosons is absolutely profound. +Consider the following superposition state, where $$\alpha$$ and $$\beta$$ are unknown: $$\begin{aligned} @@ -66,9 +78,10 @@ $$\begin{aligned} = \alpha \Ket{a}\Ket{b} + \beta \Ket{b}\Ket{a} \end{aligned}$$ -When we apply $$\hat{P}$$, we can "choose" between two "intepretations" of -its action, both shown below. Obviously, since the left-hand sides are -equal, the right-hand sides must be equal too: +When we apply $$\hat{P}$$, we can "choose" between +two "intepretations" of its action, both shown below. +Obviously, since the left-hand sides are equal, +the right-hand sides must be equal too: $$\begin{aligned} \hat{P} \Ket{\Psi(a, b)} @@ -78,25 +91,28 @@ $$\begin{aligned} &= \alpha \Ket{b}\Ket{a} + \beta \Ket{a}\Ket{b} \end{aligned}$$ -This gives us the equations $$\lambda \alpha = \beta$$ and -$$\lambda \beta = \alpha$$. In fact, just from this we could have deduced -that $$\lambda$$ can be either $$-1$$ or $$1$$. In any case, for bosons -($$\lambda = 1$$), we thus find that $$\alpha = \beta$$: +This gives us the equations $$\lambda \alpha = \beta$$ and $$\lambda \beta = \alpha$$. +In fact, just from this we could have deduced +that $$\lambda$$ can be either $$-1$$ or $$1$$. +In any case, for bosons ($$\lambda = 1$$), we thus find that $$\alpha = \beta$$: $$\begin{aligned} - \Ket{\Psi(a, b)}_b = C \big( \Ket{a}\Ket{b} + \Ket{b}\Ket{a} \big) + \Ket{\Psi(a, b)}_b + = C \big( \Ket{a}\Ket{b} + \Ket{b}\Ket{a} \big) \end{aligned}$$ -Where $$C$$ is a normalization constant. As expected, this state is -**symmetric**: switching $$a$$ and $$b$$ gives the same result. Meanwhile, for -fermions ($$\lambda = -1$$), we find that $$\alpha = -\beta$$: +Where $$C$$ is a normalization constant. +As expected, this state is **symmetric**: +switching $$a$$ and $$b$$ gives the same result. +Meanwhile, for fermions ($$\lambda = -1$$), we find that $$\alpha = -\beta$$: $$\begin{aligned} - \Ket{\Psi(a, b)}_f = C \big( \Ket{a}\Ket{b} - \Ket{b}\Ket{a} \big) + \Ket{\Psi(a, b)}_f + = C \big( \Ket{a}\Ket{b} - \Ket{b}\Ket{a} \big) \end{aligned}$$ -This state is called **antisymmetric** under exchange: switching $$a$$ and $$b$$ -causes a sign change, as we would expect for fermions. +This state is called **antisymmetric** under exchange: +switching $$a$$ and $$b$$ causes a sign change, as we would expect for fermions. Now, what if the particles $$x_1$$ and $$x_2$$ are in the same state $$a$$? For bosons, we just need to update the normalization constant $$C$$: @@ -114,7 +130,8 @@ $$\begin{aligned} = 0 \end{aligned}$$ -And this is the Pauli exclusion principle: **fermions may never -occupy the same quantum state**. One of the many notable consequences of -this is that the shells of atoms only fit a limited number of -electrons (which are fermions), since each must have a different quantum number. +And this is the Pauli exclusion principle: +**fermions may never occupy the same quantum state**. +One of the many notable consequences of this is +that the shells of atoms only fit a limited number of electrons (which are fermions), +since each must have a different quantum number. diff --git a/source/know/concept/repetition-code/index.md b/source/know/concept/repetition-code/index.md index fa039a3..ba83c1a 100644 --- a/source/know/concept/repetition-code/index.md +++ b/source/know/concept/repetition-code/index.md @@ -94,7 +94,7 @@ We could measure the state, but that would make it collapse, which is probably not what we want. The trick is to use operators called **stabilizers**, -in this case for example $$ZZI = Z_1 \otimes Z_2 \otimes I_3$$, +in this case $$ZZI = Z_1 \otimes Z_2 \otimes I_3$$, where $$I$$ is identity and $$Z$$ is the Pauli-$$Z$$ gate. The 3-qubit basis states are its eigenvectors: @@ -127,7 +127,7 @@ $$\begin{alignedat}{2} We could measure $$ZZI$$ for $$\ket{\overline{\psi}}$$, and if the eigenvalue is $$-1$$, we know that a bit flip has occurred, -whereas if the eigenvalue is $$+1$$, +but if the eigenvalue is $$+1$$, there is *maybe* no error ($$\Ket{001}$$ and $$\Ket{110}$$ are false negatives). These false negatives are fixed by including another stabilizer $$IZZ$$, @@ -170,7 +170,7 @@ thanks to the eigenvalues: | $$I$$ | $$+1$$ | $$+1$$ | | $$X_1$$ | $$-1$$ | $$+1$$ | | $$X_2$$ | $$-1$$ | $$-1$$ | -| $$X_1$$ | $$+1$$ | $$-1$$ | +| $$X_3$$ | $$+1$$ | $$-1$$ | Where e.g. $$X_3$$ denotes that the 3rd qubit was flipped. The measurement outcomes on the last three rows are called **error syndromes**, @@ -309,6 +309,9 @@ $$\begin{aligned} III \: XXX \: XXX \end{aligned}$$ +In this way, we are protected against all single-qubit errors, +but at a significant physical cost. + ## References diff --git a/source/know/concept/second-quantization/index.md b/source/know/concept/second-quantization/index.md index e446557..605ffd1 100644 --- a/source/know/concept/second-quantization/index.md +++ b/source/know/concept/second-quantization/index.md @@ -15,29 +15,26 @@ whether it is fermions or bosons that are being considered (see [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/)). Regardless of whether the system is fermionic or bosonic, -the idea is to change basis to a set of certain many-particle wave functions, -known as the **Fock states**, which are specific members of a **Fock space**, -a special kind of [Hilbert space](/know/concept/hilbert-space/), +the idea is to change basis to a set of many-particle wavefunctions +known as the **Fock states**, which are specific members of a **Fock space** +(a special kind of [Hilbert space](/know/concept/hilbert-space/)) with a well-defined number of particles. For a set of $$N$$ single-particle energy eigenstates -$$\psi_n(x)$$ and $$N$$ identical particles $$x_n$$, the Fock states are -all the wave functions which contain $$n$$ particles, for $$n$$ going from $$0$$ to $$N$$. - -So for $$n = 0$$, there is one basis vector with $$0$$ particles, -for $$n = 1$$, there are $$N$$ basis vectors with $$1$$ particle each, -for $$n = 2$$, there are $$N (N \!-\! 1)$$ basis vectors with $$2$$ particles, -etc. +$$\psi_k(x)$$ and $$N$$ identical particles $$x_k$$, +the Fock states are all the wavefunctions which contain $$n$$ particles, +for $$n$$ going from $$0$$ to $$N$$. In this basis, we define the **particle creation operators** and **particle annihilation operators**, which respectively add/remove a particle to/from a given state. -In other words, these operators relate the Fock basis vectors +In other words, these operators relate the Fock basis states to one another, and are very useful. -The point is to express the system's state in such a way that the -fermionic/bosonic constraints are automatically satisfied, and the -formulae look the same regardless of the number of particles. +The idea is to express states in such a way +that the fermionic/bosonic constraints are automatically satisfied, +and that the formulas look the same regardless of the number of particles. + ## Fermions @@ -56,6 +53,8 @@ $$\begin{aligned} \\ n &= 2: \qquad \Ket{1, 1, 0, ...} \quad \Ket{1, 0, 1, ...} \quad \Ket{0, 1, 1, ...} \quad \cdots + \\ + &\:\:\vdots \qquad \qquad \qquad \vdots \end{aligned} } \end{aligned}$$ @@ -79,16 +78,17 @@ $$\begin{aligned} The creation operator $$\hat{c}_\alpha^\dagger$$ and annihilation operator $$\hat{c}_\alpha$$ are defined to live up to their name: -they create or destroy a particle in the state $$\psi_\alpha$$: +they create or destroy a particle in the state $$\psi_\alpha$$. +Formally, this means: $$\begin{aligned} \boxed{ \begin{aligned} - \hat{c}_\alpha^\dagger \Ket{... (N_\alpha\!=\!0) ...} - &= J_\alpha \Ket{... (N_\alpha\!=\!1) ...} + \hat{c}_\alpha^\dagger \Ket{...0_\alpha...} + &= J_\alpha \Ket{...1_\alpha...} \\ - \hat{c}_\alpha \Ket{... (N_\alpha\!=\!1) ...} - &= J_\alpha \Ket{... (N_\alpha\!=\!0) ...} + \hat{c}_\alpha \Ket{...1_\alpha...} + &= J_\alpha \Ket{...0_\alpha...} \end{aligned} } \end{aligned}$$ @@ -98,7 +98,8 @@ and is necessary here to enforce the fermionic antisymmetry, when creating or destroying a particle in the $$\alpha$$th state: $$\begin{aligned} - J_\alpha = (-1)^{\sum_{j < \alpha} N_j} + J_\alpha + = (-1)^{\sum_{j < \alpha} N_j} \end{aligned}$$ So, for example, when creating a particle in state 4 @@ -110,7 +111,8 @@ $$\begin{aligned} \end{aligned}$$ The point of the Jordan-Wigner string -is that the order matters when applying the creation and annihilation operators: +is that the order matters when applying the creation and annihilation operators, +so, for example: $$\begin{aligned} \hat{c}_1^\dagger \hat{c}_2 \Ket{0, 1} @@ -124,14 +126,21 @@ $$\begin{aligned} In other words, $$\hat{c}_1^\dagger \hat{c}_2 = - \hat{c}_2 \hat{c}_1^\dagger$$, meaning that the anticommutator $$\{\hat{c}_2, \hat{c}_1^\dagger\} = 0$$. -You can verify for youself that +You can verify for yourself that the general anticommutators of these operators are given by: $$\begin{aligned} \boxed{ - \{\hat{c}_\alpha, \hat{c}_\beta\} = \{\hat{c}_\alpha^\dagger, \hat{c}_\beta^\dagger\} = 0 - \qquad \quad - \{\hat{c}_\alpha, \hat{c}_\beta^\dagger\} = \delta_{\alpha\beta} + \begin{aligned} + \{\hat{c}_\alpha, \hat{c}_\beta\} + &= 0 + \\ + \{\hat{c}_\alpha^\dagger, \hat{c}_\beta^\dagger\} + &= 0 + \\ + \{\hat{c}_\alpha, \hat{c}_\beta^\dagger\} + &= \delta_{\alpha\beta} + \end{aligned} } \end{aligned}$$ @@ -141,24 +150,29 @@ Note that these are *scalar* zeros: $$\begin{aligned} \boxed{ - \hat{c}_\alpha^\dagger \Ket{... (N_\alpha\!=\!1) ...} = 0 - \qquad \quad - \hat{c}_\alpha \Ket{... (N_\alpha\!=\!0) ...} = 0 + \begin{aligned} + \hat{c}_\alpha^\dagger \Ket{...1_\alpha...} + &= 0 + \\ + \hat{c}_\alpha \Ket{...0_\alpha...} + &= 0 + \end{aligned} } \end{aligned}$$ Finally, as has already been suggested by the notation, they are each other's adjoint: $$\begin{aligned} - \matrixel{... (N_\alpha\!=\!1) ...}{\hat{c}_\alpha^\dagger}{... (N_\alpha\!=\!0) ...} - = \matrixel{...(N_\alpha\!=\!0) ...}{\hat{c}_\alpha}{... (N_\alpha\!=\!1) ...} + \matrixel{...1_\alpha...}{\hat{c}_\alpha^\dagger}{...0_\alpha...} + = \matrixel{...0_\alpha...}{\hat{c}_\alpha}{...1_\alpha...}^{*} \end{aligned}$$ Let us now use these operators to define the **number operator** $$\hat{N}_\alpha$$ as follows: $$\begin{aligned} \boxed{ - \hat{N}_\alpha = \hat{c}_\alpha^\dagger \hat{c}_\alpha + \hat{N}_\alpha + = \hat{c}_\alpha^\dagger \hat{c}_\alpha } \end{aligned}$$ @@ -171,6 +185,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Bosons Bosons do not need to obey the Pauli exclusion principle, so multiple can occupy a single state. @@ -188,8 +203,10 @@ $$\begin{aligned} n &= 2: \qquad \Ket{1, 1, 0, ...} \quad \Ket{1, 0, 1, ...} \quad \Ket{0, 1, 1, ...} \quad \cdots \\ - &\qquad\:\:\: + &\qquad\:\,\, \qquad \Ket{2, 0, 0, ...} \quad \Ket{0, 2, 0, ...} \quad \Ket{0, 0, 2, ...} \quad \cdots + \\ + &\:\:\vdots \qquad \qquad \qquad \vdots \end{aligned} } \end{aligned}$$ @@ -212,23 +229,31 @@ $$\begin{gathered} \end{aligned} }\end{gathered}$$ -Applying the annihilation operator $$\hat{c}_\alpha$$ when there are zero -particles in $$\alpha$$ will quench the state: +Applying the annihilation operator $$\hat{c}_\alpha$$ +when there are zero particles in $$\alpha$$ quenches the state: $$\begin{aligned} \boxed{ - \hat{c}_\alpha \Ket{... (N_\alpha\!=\!0) ...} = 0 + \hat{c}_\alpha \Ket{...0_\alpha...} + = 0 } \end{aligned}$$ There is no Jordan-Wigner string, and therefore no sign change when commuting. -Consequently, these operators therefore satisfy the following: +Consequently, these operators satisfy the following commutators: $$\begin{aligned} \boxed{ - [\hat{c}_\alpha, \hat{c}_\beta] = [\hat{c}_\alpha^\dagger, \hat{c}_\beta^\dagger] = 0 - \qquad - [\hat{c}_\alpha, \hat{c}_\beta^\dagger] = \delta_{\alpha\beta} + \begin{aligned} + [\hat{c}_\alpha, \hat{c}_\beta] + &= 0 + \\ + [\hat{c}_\alpha^\dagger, \hat{c}_\beta^\dagger] + &= 0 + \\ + [\hat{c}_\alpha, \hat{c}_\beta^\dagger] + &= \delta_{\alpha\beta} + \end{aligned} } \end{aligned}$$ @@ -237,90 +262,93 @@ ensure that $$\hat{N}_\alpha$$ keeps the same nice form: $$\begin{aligned} \boxed{ - \hat{N}_\alpha = \hat{c}_\alpha^\dagger \hat{c}_\alpha + \hat{N}_\alpha + = \hat{c}_\alpha^\dagger \hat{c}_\alpha } \end{aligned}$$ + ## Operators -Traditionally, an operator $$\hat{V}$$ simultaneously acting on $$N$$ indentical particles -is the sum of the individual single-particle operators $$\hat{V}_1$$ acting on the $$n$$th particle: +In the second quantization, +changing between different bases of single-particle states +is done in the usual way, where $$\alpha$$ and $$b$$ need not be in the same basis. +Note that $$\Ket{0}$$ is the zero-particle Fock state, +and $$\Ket{\alpha}$$ etc. are one-particle Fock states: $$\begin{aligned} - \hat{V} - = \sum_{n = 1}^N \hat{V}_1 + \hat{c}_b^\dagger \Ket{0} + = \Ket{b} + = \sum_{\alpha} \Ket{\alpha} \inprod{\alpha}{b} + = \sum_{\alpha} \inprod{\alpha}{b} \hat{c}_\alpha^\dagger \Ket{0} \end{aligned}$$ -This can be rewritten using the second quantization operators as follows: +With this, we define the **field operators**, +which create or destroy a particle at a position $$\vb{r}$$: $$\begin{aligned} \boxed{ - \hat{V} - = \sum_{\alpha, \beta} \matrixel{\alpha}{\hat{V}_1}{\beta} \hat{c}_\alpha^\dagger \hat{c}_\beta + \hat{\Psi}^\dagger(\vb{r}) + = \sum_{\alpha} \inprod{\alpha}{\vb{r}} \hat{c}_\alpha^\dagger + \qquad \qquad + \hat{\Psi}(\vb{r}) + = \sum_{\alpha} \inprod{\vb{r}}{\alpha} \hat{c}_\alpha } \end{aligned}$$ -Where the matrix element $$\matrixel{\alpha}{\hat{V}_1}{\beta}$$ is to be -evaluated in the normal way: - -$$\begin{aligned} - \matrixel{\alpha}{\hat{V}_1}{\beta} - = \int \psi_\alpha^*(\vec{r}) \: \hat{V}_1(\vec{r}) \: \psi_\beta(\vec{r}) \dd{\vec{r}} -\end{aligned}$$ - -Similarly, given some two-particle operator $$\hat{V}$$ in first-quantized form: +By the same basis-changing principle, +any single-particle (non-interacting) operator $$\hat{V}$$ can be translated +to its second-quantized $$N$$-particle version as follows: $$\begin{aligned} \hat{V} - = \sum_{n \neq m} v(\vec{r}_n, \vec{r}_m) + &= \sum_{\alpha, \beta} \ket{\alpha} \matrixel{\alpha}{\hat{V}}{\beta} \bra{\beta} + = \sum_{\alpha, \beta} \ket{\hat{c}_\alpha^\dagger 0} \matrixel{\alpha}{\hat{V}}{\beta} \bra{\hat{c}_\beta^\dagger 0} \end{aligned}$$ -We can rewrite this in second-quantized form as follows. -Note the ordering of the subscripts: +We take out the creation operators, +which allows us to generalize to multi-particle states: $$\begin{aligned} \boxed{ \hat{V} - = \sum_{\alpha, \beta, \gamma, \delta} - v_{\alpha \beta \gamma \delta} \hat{c}_\alpha^\dagger \hat{c}_\beta^\dagger \hat{c}_\delta \hat{c}_\gamma + = \sum_{\alpha, \beta} \matrixel{\alpha}{\hat{V}}{\beta} \hat{c}_\alpha^\dagger \hat{c}_\beta } \end{aligned}$$ -Where the constant $$v_{\alpha \beta \gamma \delta}$$ is defined from the -single-particle wave functions: +Where the matrix element $$\matrixel{\alpha}{\hat{V}}{\beta}$$ +is to be evaluated in the normal way: $$\begin{aligned} - v_{\alpha \beta \gamma \delta} - = \iint \psi_\alpha^*(\vec{r}_1) \: \psi_\beta^*(\vec{r}_2) - \: v(\vec{r}_1, \vec{r}_2) \: \psi_\gamma(\vec{r}_1) - \: \psi_\delta(\vec{r}_2) \dd{\vec{r}_1} \dd{\vec{r}_2} + \matrixel{\alpha}{\hat{V}}{\beta} + = \int \psi_\alpha^*(\vb{r}) \: \hat{V}(\vb{r}) \: \psi_\beta(\vb{r}) \dd{\vb{r}} \end{aligned}$$ -Finally, in the second quantization, changing basis is done in the usual way: +In the same way, a two-particle interaction operator $$\hat{W}$$ +can be rewritten in the form below. +Note the ordering of the operators' subscripts: $$\begin{aligned} - \hat{c}_b^\dagger \Ket{0} - = \Ket{b} - = \sum_{\alpha} \Ket{\alpha} \Inprod{\alpha}{b} - = \sum_{\alpha} \Inprod{\alpha}{b} \hat{c}_\alpha^\dagger \Ket{0} + \boxed{ + \hat{W} + = \sum_{\alpha, \beta, \gamma, \delta} + W_{\alpha \beta \gamma \delta} \: \hat{c}_\alpha^\dagger \hat{c}_\beta^\dagger \hat{c}_\delta \hat{c}_\gamma + } \end{aligned}$$ -Where $$\alpha$$ and $$b$$ need not be in the same basis. -With this, we can define the **field operators**, -which create or destroy a particle at a given position $$\vec{r}$$: +Where the constant $$W_{\alpha \beta \gamma \delta}$$ +is defined from the single-particle wavefunctions like so: $$\begin{aligned} - \boxed{ - \hat{\Psi}^\dagger(\vec{r}) - = \sum_{\alpha} \Inprod{\alpha}{\vec{r}} \hat{c}_\alpha^\dagger - \qquad \quad - \hat{\Psi}(\vec{r}) - = \sum_{\alpha} \Inprod{\vec{r}}{\alpha} \hat{c}_\alpha - } + W_{\alpha \beta \gamma \delta} + \equiv \iint \psi_\alpha^*(\vb{r}_1) \: \psi_\beta^*(\vb{r}_2) + \: W(\vb{r}_1, \vb{r}_2) \: \psi_\gamma(\vb{r}_1) + \: \psi_\delta(\vb{r}_2) \dd{\vb{r}_1} \dd{\vb{r}_2} \end{aligned}$$ + ## References 1. L.E. Ballentine, *Quantum mechanics: a modern development*, 2nd edition, diff --git a/source/know/concept/shors-algorithm/index.md b/source/know/concept/shors-algorithm/index.md index 678d5d2..bab88a3 100644 --- a/source/know/concept/shors-algorithm/index.md +++ b/source/know/concept/shors-algorithm/index.md @@ -29,10 +29,6 @@ This is a so-called *hidden subgroup problem* for a *finite Abelian group*. With minimal modifications, Shor's algorithm can solve practically every such problem. - - -## Integer factorization - Originally, Shor's algorithm was designed to factorize an integer $$N$$. For reasons explained later, this means our goal is to find the period $$s$$ of @@ -40,7 +36,7 @@ the modular exponentiation function $$f$$: $$\begin{aligned} f(x) - = a^x \bmod N + \equiv a^x \bmod N \end{aligned}$$ For a given $$a$$ and $$N$$. @@ -72,7 +68,8 @@ $$\begin{aligned} = \frac{1}{\sqrt{Q}} \sum_{x = 0}^{Q - 1} \Ket{x} \Ket{0}^{\otimes q} \end{aligned}$$ -Where $$Q = 2^q$$, and $$\Ket{x}$$ is the computational basis state $$\Ket{x_1} \cdots \Ket{x_q}$$. +Where $$Q \equiv 2^q$$, +and $$\Ket{x}$$ is the computational basis state $$\Ket{x_1} \cdots \Ket{x_q}$$. Moving on to $$U_f$$: $$\begin{aligned} @@ -81,32 +78,41 @@ $$\begin{aligned} \frac{1}{\sqrt{Q}} \sum_{x = 0}^{Q - 1} \Ket{x} \Ket{f(x)} \end{aligned}$$ -Then we measure $$f(x)$$, causing it collapse as follows -for an unknown arbitrary value of $$x_0$$: +Then we measure $$f(x)$$, causing it collapse +for an unknown arbitrary value of $$x_0$$. +Let $$L$$ be number of periods that fit in the available qubits, +then we know that: $$\begin{aligned} - f(x_0) = f(x_0 + s) = f(x_0 + 2s) = \cdots = f(x_0 + (L-1) s) + f(x_0) + = f(x_0 + s) + = \cdots + = f(x_0 + \ell s) + = \cdots + = f(x_0 + (L-1) s) \end{aligned}$$ Due to [entanglement](/know/concept/quantum-entanglement/), -the unmeasured (top $$q$$) qubits change state into a superposition: +the unmeasured (top $$q$$) qubits change state, so we now have: $$\begin{aligned} - \frac{1}{\sqrt{L}} \sum_{\ell = 0}^{L - 1} \Ket{x_0 + \ell s} + \frac{1}{\sqrt{L}} \sum_{\ell = 0}^{L - 1} \Ket{x_0 + \ell s} \Ket{f(x_0)} \end{aligned}$$ Clearly, there is a periodic structure here, but we cannot measure it directly, -because we do not know the value of $$x_0$$, -which, to make matters worse, changes every time we run the algorithm. -This is where the QFT comes in, which outputs the following state: +since we do not know the value of $$x_0$$, +which, to make matters worse, changes every time we run the algorithm! +But now we apply the QFT, which outputs the state below, +where $$\omega_Q$$ is a $$Q$$th root of unity. +From now on, we no longer write the measured qubits $$\Ket{f(x_0)}$$, leaving: $$\begin{aligned} \frac{1}{\sqrt{QL}} \sum_{k = 0}^{Q - 1} \bigg( \sum_{\ell = 0}^{L - 1} \omega_Q^{(x_0 + \ell s) k} \bigg) \Ket{k} \end{aligned}$$ -Where $$\omega_Q$$ is a $$Q$$th root of unity. -Measuring this state yields a $$\Ket{k}$$, with a probability $$P(k)$$: +Measuring this state causes a collapse into some $$\Ket{k}$$ +with a probability $$P(k)$$: $$\begin{aligned} P(k) @@ -116,9 +122,9 @@ $$\begin{aligned} \end{aligned}$$ The last step holds because $$|\omega_Q| = 1$$. -Surprisingly, this implies that we did not need -to perform the measurement of $$f(x)$$ earlier! -This makes sense: the period $$s$$ does not depend on $$x_0$$, +Surprisingly, $$x_0$$ has disappeared, +implying that we did not need to perform the measurement of $$f(x)$$ earlier! +This makes some sense: the period $$s$$ does not depend on $$x_0$$, so why would we need an implicit $$x_0$$ to determine $$s$$? So, what does the above probability $$P(k)$$ work out to? @@ -134,12 +140,13 @@ $$\begin{alignedat}{2} Where the latter case was evaluated as a geometric series. The condition $$\omega_Q^{sk}\!=\!1$$ is equivalent to asking -if $$sk$$ is a multiple of $$Q$$, i.e. if $$sk = cQ$$, for an integer $$c$$. +if $$sk$$ is a multiple of $$Q$$. +i.e. if $$sk = cQ$$, for an integer $$c$$. Recall that $$L$$ is the number of times that $$s$$ fits in $$Q$$, so $$L\!=\!\lfloor Q / s \rfloor$$. -Assuming $$Q/s$$ is an integer, then $$L\!=\!Q/s$$ and $$Q\!=\!s L$$, -which tells us that +For now, let us assume that $$Q/s$$ is an integer, +then $$L\!=\!Q/s$$ and $$Q\!=\!s L$$, which tells us that $$\omega_Q^{sk}\!=\!\omega_{s L}^{s k}\!=\!\omega_L^k$$. This implies that if $$k$$ is a multiple of $$L$$ (i.e. $$k\!=\!c L$$), then $$\omega_L^k\!=\!1$$, so $$P(k) = L / Q$$, @@ -245,9 +252,11 @@ $$\begin{aligned} = a^x \bmod N \end{aligned}$$ -$$N$$ is the number to factorize, and $$a$$ is a random integer *coprime* to $$N$$, +$$N$$ is the number to factorize, and $$a$$ is an integer that we can choose. +For this to work, we should pick an $$a$$ that is *coprime* to $$N$$, meaning $$\gcd(a, N) = 1$$. -The fact that $$s$$ is the period of $$f$$ for a certain $$a$$-value, implies that: +Then the fact that $$s$$ is the period of $$f$$ +for a certain $$a$$-value implies that: $$\begin{aligned} a^x @@ -257,7 +266,7 @@ $$\begin{aligned} = a^s \bmod N \end{aligned}$$ -Suppose that $$s$$ is even. In that case, +For now, suppose that $$s$$ is even. In that case, we can rewrite the above equation as follows: $$\begin{aligned} @@ -273,7 +282,8 @@ $$\begin{aligned} = 0 \bmod N \end{aligned}$$ -Because $$s$$ is even by assumption, the two factors on the left are integers, +Because $$s$$ is even by assumption (for now), +the two factors on the left are integers, and as just mentioned, their product is a multiple of $$N$$. Then we only need to calculate: diff --git a/source/know/concept/sokhotski-plemelj-theorem/index.md b/source/know/concept/sokhotski-plemelj-theorem/index.md index 445b029..e139954 100644 --- a/source/know/concept/sokhotski-plemelj-theorem/index.md +++ b/source/know/concept/sokhotski-plemelj-theorem/index.md @@ -10,7 +10,7 @@ layout: "concept" --- The goal is to evaluate integrals of the following form, -where $$f(x)$$ is assumed to be continuous in the integration interval $$[a, b]$$: +where $$f(x)$$ is real and continuous in the integration interval $$[a, b]$$: $$\begin{aligned} \lim_{\eta \to 0^+} \int_a^b \frac{f(x)}{x + i \eta} \dd{x} @@ -56,7 +56,7 @@ $$\begin{aligned} &= \lim_{m \to +\infty} \frac{\pi}{\pi} \int_a^b \frac{m}{1 + m^2 x^2} f(x) \dd{x} \end{aligned}$$ -The expression $$m / \pi (1 + m^2 x^2)$$ is a so-called *nascent delta function*, +The expression $$m / (\pi (1 + m^2 x^2))$$ is a so-called *nascent delta function*, meaning that in the limit $$m \to +\infty$$ it converges to the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(x)$$: diff --git a/source/know/concept/thermodynamic-potential/index.md b/source/know/concept/thermodynamic-potential/index.md index b15c011..60eee78 100644 --- a/source/know/concept/thermodynamic-potential/index.md +++ b/source/know/concept/thermodynamic-potential/index.md @@ -12,17 +12,17 @@ layout: "concept" whose minima or maxima represent equilibrium states of a system. Such functions are either energies (hence *potential*) or entropies. -Which potential (of many) decides the equilibrium states for a given system? -That depends which variables are assumed to already be in automatic equilibrium. -Such variables are known as the **natural variables** of that potential. -For example, if a system can freely exchange heat with its environment, -and is consequently assumed to be at the same temperature $$T = T_{\mathrm{env}}$$, +Of the many options, which potential decides the equilibrium state for a given system? +It depends on which variables are assumed to be in automatic equilibrium. +Such variables are called the **natural variables** of that potential. +For example, if a system can exchange heat with its environment, +and is consequently at the same temperature $$T = T_{\mathrm{env}}$$, then $$T$$ must be a natural variable. The link from natural variables to potentials is established by [thermodynamic ensembles](/know/category/thermodynamic-ensembles/). -Once enough natural variables have been found, +Once the natural variables have been determined, the appropriate potential can be selected from the list below. All non-natural variables can then be calculated by taking partial derivatives of the potential @@ -48,8 +48,8 @@ $$\begin{aligned} \end{aligned}$$ It is a function of the entropy $$S$$, volume $$V$$, and particle count $$N$$: -these are its natural variables. -An infinitesimal change $$\dd{U}$$ is as follows: +these are its natural variables, +so an infinitesimal change $$\dd{U}$$ is as follows: $$\begin{aligned} \boxed{ @@ -59,7 +59,7 @@ $$\begin{aligned} The non-natural variables are temperature $$T$$, pressure $$P$$, and chemical potential $$\mu$$. -They can be recovered by differentiating $$U$$ +These can be recovered by differentiating $$U$$ with respect to the natural variables $$S$$, $$V$$, and $$N$$: $$\begin{aligned} @@ -92,8 +92,8 @@ $$\begin{aligned} \end{aligned}$$ It is a function of the entropy $$S$$, pressure $$P$$, and particle count $$N$$: -these are its natural variables. -An infinitesimal change $$\dd{H}$$ is as follows: +these are its natural variables, +so an infinitesimal change $$\dd{H}$$ is as follows: $$\begin{aligned} \boxed{ @@ -103,7 +103,7 @@ $$\begin{aligned} The non-natural variables are temperature $$T$$, volume $$V$$, and chemical potential $$\mu$$. -They can be recovered by differentiating $$H$$ +These can be recovered by differentiating $$H$$ with respect to the natural variables $$S$$, $$P$$, and $$N$$: $$\begin{aligned} @@ -132,8 +132,8 @@ $$\begin{aligned} \end{aligned}$$ It depends on the temperature $$T$$, volume $$V$$, and particle count $$N$$: -these are natural variables. -An infinitesimal change $$\dd{H}$$ is as follows: +these are its natural variables, +so an infinitesimal change $$\dd{H}$$ is as follows: $$\begin{aligned} \boxed{ @@ -142,8 +142,8 @@ $$\begin{aligned} \end{aligned}$$ The non-natural variables are -entropy $$S$$, pressure $$P$$, and chemical potential $$\mu$$. -They can be recovered by differentiating $$F$$ +the entropy $$S$$, pressure $$P$$, and chemical potential $$\mu$$. +These can be recovered by differentiating $$F$$ with respect to the natural variables $$T$$, $$V$$, and $$N$$: $$\begin{aligned} @@ -171,8 +171,8 @@ $$\begin{aligned} \end{aligned}$$ It depends on the temperature $$T$$, pressure $$P$$, and particle count $$N$$: -they are natural variables. -An infinitesimal change $$\dd{G}$$ is as follows: +they are its natural variables, +so an infinitesimal change $$\dd{G}$$ is as follows: $$\begin{aligned} \boxed{ @@ -181,7 +181,7 @@ $$\begin{aligned} \end{aligned}$$ The non-natural variables are -entropy $$S$$, volume $$V$$, and chemical potential $$\mu$$. +the entropy $$S$$, volume $$V$$, and chemical potential $$\mu$$. These can be recovered by differentiating $$G$$ with respect to the natural variables $$T$$, $$P$$, and $$N$$: @@ -210,8 +210,8 @@ $$\begin{aligned} \end{aligned}$$ It depends on temperature $$T$$, volume $$V$$, and chemical potential $$\mu$$: -these are natural variables. -An infinitesimal change $$\dd{\Omega}$$ is as follows: +these are its natural variables, +so an infinitesimal change $$\dd{\Omega}$$ is as follows: $$\begin{aligned} \boxed{ @@ -239,7 +239,8 @@ $$\begin{aligned} ## Entropy The **entropy** $$S$$, in units of energy over temperature, -is an odd duck, but nevertheless used as a thermodynamic potential. +is an odd duck, but nevertheless used as a thermodynamic potential, +to be maximized instead of minimized. It is given by: $$\begin{aligned} @@ -249,8 +250,8 @@ $$\begin{aligned} \end{aligned}$$ It depends on the internal energy $$U$$, volume $$V$$, and particle count $$N$$: -they are natural variables. -An infinitesimal change $$\dd{S}$$ is as follows: +they are its natural variables, +so an infinitesimal change $$\dd{S}$$ is as follows: $$\begin{aligned} \boxed{ diff --git a/source/know/concept/wkb-approximation/index.md b/source/know/concept/wkb-approximation/index.md index ef57a3b..fb04414 100644 --- a/source/know/concept/wkb-approximation/index.md +++ b/source/know/concept/wkb-approximation/index.md @@ -8,24 +8,25 @@ categories: layout: "concept" --- -In quantum mechanics, the **Wentzel-Kramers-Brillouin** or simply the **WKB -approximation** is a technique to approximate the wave function $$\psi(x)$$ of -the one-dimensional time-independent Schrödinger equation. It is an example -of a **semiclassical approximation**, because it tries to find a -balance between classical and quantum physics. +In quantum mechanics, the **Wentzel-Kramers-Brillouin** +or simply the **WKB approximation** +is a technique to approximate the wavefunction $$\psi(x)$$ +of the 1D time-independent Schrödinger equation. +It is an example of a **semiclassical approximation**, +because it tries to find a balance between classical and quantum physics. In classical mechanics, a particle travelling in a potential $$V(x)$$ along a path $$x(t)$$ has a total energy $$E$$ as follows, which we rearrange: $$\begin{aligned} - E = \frac{1}{2} m \dot{x}^2 + V(x) - \quad \implies \quad + E = \frac{1}{2} m (x')^2 + V(x) + \qquad \implies \qquad m^2 (x')^2 = 2 m (E - V(x)) \end{aligned}$$ The left-hand side of the rearranged version is simply the momentum squared, -so we define the magnitude of the momentum $$p(x)$$ accordingly: +so we know that the magnitude of the momentum $$p(x)$$ is: $$\begin{aligned} p(x) = \sqrt{2 m (E - V(x))} @@ -38,8 +39,9 @@ We rewrite the Schrödinger equation: $$\begin{aligned} 0 - = \dvn{2}{\psi}{x} + \frac{2 m}{\hbar^2} (E - V) \psi - = \dvn{2}{\psi}{x} + \frac{p^2}{\hbar^2} \psi + &= \dvn{2}{\psi}{x} + \frac{2 m}{\hbar^2} (E - V) \psi + \\ + &= \dvn{2}{\psi}{x} + \frac{p^2}{\hbar^2} \psi \end{aligned}$$ If $$V(x)$$ were constant, and by extension $$p(x)$$ too, then the solution @@ -50,20 +52,20 @@ $$\begin{aligned} = \psi(0) \exp(\pm i p x / \hbar) \end{aligned}$$ -This form is reminiscent of the generator of translations. In practice, -$$V(x)$$ and $$p(x)$$ vary with $$x$$, but we can still salvage this solution +In practice, $$V(x)$$ and $$p(x)$$ vary with $$x$$, +but we can still salvage this solution by assuming that $$V(x)$$ varies slowly compared to the wavelength -$$\lambda(x) = 2 \pi / k(x)$$, where $$k(x) = p(x) / \hbar$$ is the -wavenumber. The solution then takes the following form: +$$2 \pi / k(x)$$, where $$k(x) = p(x) / \hbar$$ is the wavenumber. +The solution then takes the following form: $$\begin{aligned} \psi(x) = \psi(0) \exp\!\Big(\!\pm\! \frac{i}{\hbar} \int_0^x \chi(\xi) \dd{\xi} \Big) \end{aligned}$$ -$$\chi(\xi)$$ is an unknown function, which intuitively should be related -to $$p(x)$$. The purpose of the integral is to accumulate the change of -$$\chi$$ from the initial point $$0$$ to the current position $$x$$. +$$\chi(\xi)$$ is an unknown function, which intuitively should be related to $$p(x)$$. +The purpose of the integral is to accumulate the change of $$\chi$$ +from the initial point $$0$$ to the current position $$x$$. Let us write this as an indefinite integral for convenience: $$\begin{aligned} @@ -71,111 +73,118 @@ $$\begin{aligned} = \psi(0) \exp\!\bigg( \!\pm\! \frac{i}{\hbar} \Big( \int \chi(x) \dd{x} - C \Big) \bigg) \end{aligned}$$ -Where $$C = \int \chi(x) \dd{x} |_{x = 0}$$ is the initial point of the definite integral. +Where $$C = \int \chi(x) \dd{x} |_{x = 0}$$ is +the initial point of the definite integral. For simplicity, we absorb the constant $$C$$ into $$\psi(0)$$. We can now clearly see that: $$\begin{aligned} - \psi'(x) = \pm \frac{i}{\hbar} \chi(x) \psi(x) - \quad \implies \quad - \chi(x) = \pm \frac{\hbar}{i} \frac{\psi'(x)}{\psi(x)} + \psi'(x) + = \pm \frac{i}{\hbar} \chi(x) \psi(x) \end{aligned}$$ -Next, we insert this ansatz for $$\psi(x)$$ into the Schrödinger equation -to get: +We insert this ansatz for $$\psi(x)$$ into the Schrödinger equation to get: $$\begin{aligned} 0 &= \pm \frac{i}{\hbar} \dv{(\chi \psi)}{x} + \frac{p^2}{\hbar^2} \psi - = \pm \frac{i}{\hbar} \chi' \psi \pm \frac{i}{\hbar} \chi \psi' + \frac{p^2}{\hbar^2} \psi - = \pm \frac{i}{\hbar} \chi' \psi - \frac{1}{\hbar^2} \chi^2 \psi + \frac{p^2}{\hbar^2} \psi + \\ + &= \pm \frac{i}{\hbar} \chi' \psi \pm \frac{i}{\hbar} \chi \psi' + \frac{p^2}{\hbar^2} \psi + \\ + &= \pm \frac{i}{\hbar} \chi' \psi - \frac{1}{\hbar^2} \chi^2 \psi + \frac{p^2}{\hbar^2} \psi \end{aligned}$$ -Dividing out $$\psi$$ and rearranging gives us the following, which is -still exact: +Dividing out $$\psi$$ and rearranging gives us the following, which is still exact: $$\begin{aligned} \pm \frac{\hbar}{i} \chi' = p^2 - \chi^2 \end{aligned}$$ -Next, we expand this as a power series of $$\hbar$$. This is why it is -called *semiclassical*: so far we have been using full quantum mechanics, -but now we are treating $$\hbar$$ as a parameter which controls the -strength of quantum effects: +Next, we expand this as a power series of $$\hbar$$. +This is why it is called *semiclassical*: +so far we have been using full quantum mechanics, +but now we are treating $$\hbar$$ as a parameter +which controls the strength of quantum effects: $$\begin{aligned} - \chi(x) = \chi_0(x) + \frac{\hbar}{i} \chi_1(x) + \frac{\hbar^2}{i^2} \chi_2(x) + ... + \chi(x) + = \chi_0(x) + \frac{\hbar}{i} \chi_1(x) + \frac{\hbar^2}{i^2} \chi_2(x) + \cdots +\end{aligned}$$ + +The heart of the WKB approximation is its assumption that quantum effects +are sufficiently weak that we only need to consider +the first two terms of this expansion, +i.e. $$\hbar^2$$ is so small that it is negligible. +Therefore, our approximated wavefunction $$\psi(x)$$ now looks like this: + +$$\begin{aligned} + \psi(x) + &\approx \psi(0) \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int \chi_0(x) \dd{x} \Big) \exp\!\Big( \!\pm\! \int \chi_1(x) \dd{x} \Big) \end{aligned}$$ -The heart of the WKB approximation is its assumption that quantum effects are -sufficiently weak (i.e. $$\hbar$$ is small enough) that we only need to -consider the first two terms, or, more specifically, that we only go up to -$$\hbar$$, not $$\hbar^2$$ or higher. Inserting the first two terms of this -expansion into the equation: +Inserting the expansion's first two terms into our equation for $$\chi(x)$$ gives: $$\begin{aligned} \pm \frac{\hbar}{i} \chi_0' &= p^2 - \chi_0^2 - 2 \frac{\hbar}{i} \chi_0 \chi_1 \end{aligned}$$ -Where we have discarded all terms containing $$\hbar^2$$. At order -$$\hbar^0$$, we then get the expected classical result for $$\chi_0(x)$$: +Where we have discarded all terms containing $$\hbar^2$$. +At order $$\hbar^0$$, we then get the expected classical result for $$\chi_0(x)$$: $$\begin{aligned} 0 = p^2 - \chi_0^2 - \quad \implies \quad - \chi_0(x) = p(x) + \qquad \implies \qquad + \chi_0(x) + = p(x) \end{aligned}$$ -While at order $$\hbar$$, we get the following quantum-mechanical -correction: +While at order $$\hbar$$, we get the following quantum-mechanical correction: $$\begin{aligned} \pm \frac{\hbar}{i} \chi_0' = - 2 \frac{\hbar}{i} \chi_0 \chi_1 - \quad \implies \quad - \chi_1(x) = \mp \frac{1}{2} \frac{\chi_0'(x)}{\chi_0(x)} + \qquad \implies \qquad + \chi_1(x) + = \mp \frac{1}{2} \frac{\chi_0'(x)}{\chi_0(x)} \end{aligned}$$ -Therefore, our approximated wave function $$\psi(x)$$ currently looks like -this: - -$$\begin{aligned} - \psi(x) - &\approx \psi(0) \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int \chi_0(x) \dd{x} \Big) \exp\!\Big( \!\pm\! \int \chi_1(x) \dd{x} \Big) -\end{aligned}$$ - -We can reduce the latter exponential using integration by substitution: +We can use this to simplify the latter exponential in $$\psi(x)$$ +using integration by substitution: $$\begin{aligned} \exp\!\Big( \!\pm\! \int \chi_1(x) \dd{x} \Big) &= \exp\!\Big( \!-\! \frac{1}{2} \int \frac{\chi_0'(x)}{\chi_0(x)} \dd{x} \Big) - = \exp\!\Big( \!-\! \frac{1}{2} \int \frac{1}{\chi_0}\:d\chi_0 \Big) + \\ + &= \exp\!\Big( \!-\! \frac{1}{2} \int \frac{1}{\chi_0}\:d\chi_0 \Big) \\ &= \exp\!\Big( \!-\! \frac{1}{2} \ln\!\big(\chi_0(x)\big) \Big) - = \frac{1}{\sqrt{\chi_0(x)}} - = \frac{1}{\sqrt{p(x)}} + \\ + &= \frac{1}{\sqrt{\chi_0(x)}} \end{aligned}$$ -In the WKB approximation for $$E > V$$, the solution $$\psi(x)$$ is thus -given by: +In the WKB approximation for $$E > V$$, +the solution $$\psi(x)$$ is therefore given by: $$\begin{aligned} \boxed{ - \psi(x) \approx \frac{A}{\sqrt{p(x)}} \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int p(x) \dd{x} \Big) + \psi(x) + \approx \frac{A}{\sqrt{p(x)}} \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int p(x) \dd{x} \Big) } \end{aligned}$$ -What if $$E < V$$? In classical mechanics, this is just not allowed; a ball -cannot simply go through a potential bump without the necessary energy. +What if $$E < V$$? In classical mechanics, this is not allowed: +a ball cannot simply go through or over a potential bump without the necessary energy. On the other hand, in quantum physics, particles can **tunnel** through barriers. Luckily, the only thing we need to change for the WKB approximation is to let the momentum take imaginary values: $$\begin{aligned} - p(x) = \sqrt{2 m (E - V(x))} = i \sqrt{2 m (V(x) - E)} + p(x) + = \sqrt{2 m (E - V(x))} + = i \sqrt{2 m (V(x) - E)} \end{aligned}$$ And then take the absolute value in the appropriate place in front of $$\psi(x)$$: @@ -186,12 +195,13 @@ $$\begin{aligned} } \end{aligned}$$ -In the classical region ($$E > V$$), the wave function oscillates, and -in the quantum-physical region ($$E < V$$) it is exponential. +In the classical region ($$E > V$$), the wavefunction oscillates, +and in the quantum-physical region ($$E < V$$) it is exponential. Note that for $$E \approx V$$ the approximation breaks down, because of the appearance of $$p(x)$$ in the denominator. + ## References 1. D.J. Griffiths, D.F. Schroeter, *Introduction to quantum mechanics*, 3rd edition, -- cgit v1.3