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Diffstat (limited to 'source')
| -rw-r--r-- | source/know/concept/bernoullis-theorem/index.md | 29 | ||||
| -rw-r--r-- | source/know/concept/central-limit-theorem/index.md | 26 | ||||
| -rw-r--r-- | source/know/concept/clausius-mossotti-relation/index.md | 27 | ||||
| -rw-r--r-- | source/know/concept/ehrenfests-theorem/index.md | 75 | ||||
| -rw-r--r-- | source/know/concept/ficks-laws/index.md | 54 | ||||
| -rw-r--r-- | source/know/concept/grad-shafranov-equation/index.md | 13 | ||||
| -rw-r--r-- | source/know/concept/holomorphic-function/index.md | 27 | ||||
| -rw-r--r-- | source/know/concept/larmor-precession/index.md | 26 | ||||
| -rw-r--r-- | source/know/concept/lubrication-theory/index.md | 67 | ||||
| -rw-r--r-- | source/know/concept/lyddane-sachs-teller-relation/index.md | 45 | ||||
| -rw-r--r-- | source/know/concept/prandtl-equations/index.md | 58 | ||||
| -rw-r--r-- | source/know/concept/probability-current/index.md | 66 | ||||
| -rw-r--r-- | source/know/concept/rabi-oscillation/index.md | 37 | ||||
| -rw-r--r-- | source/know/concept/time-dependent-perturbation-theory/index.md | 4 |
14 files changed, 277 insertions, 277 deletions
diff --git a/source/know/concept/bernoullis-theorem/index.md b/source/know/concept/bernoullis-theorem/index.md index 867c443..2795f22 100644 --- a/source/know/concept/bernoullis-theorem/index.md +++ b/source/know/concept/bernoullis-theorem/index.md @@ -12,36 +12,21 @@ layout: "concept" For inviscid fluids, **Bernouilli's theorem** states that an increase in flow velocity $$\va{v}$$ is paired with a decrease in pressure $$p$$ and/or potential energy. -For a qualitative argument, look no further than -one of the [Euler equations](/know/concept/euler-equations/), -with a [material derivative](/know/concept/material-derivative/): - -$$\begin{aligned} - \frac{\mathrm{D} \va{v}}{\mathrm{D} t} - = \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v} - = \va{g} - \frac{\nabla p}{\rho} -\end{aligned}$$ - -Assuming that $$\va{v}$$ is constant in $$t$$, -it becomes clear that a higher $$\va{v}$$ requires a lower $$p$$. - - -## Simple form - -For an incompressible fluid +Quantitatively, for an incompressible fluid with a time-independent velocity field $$\va{v}$$ (i.e. **steady flow**), -Bernoulli's theorem formally states that the -**Bernoulli head** $$H$$ is constant along a streamline: +Bernoulli's theorem states that +the **Bernoulli head** $$H$$ is constant along every streamline: $$\begin{aligned} \boxed{ H - = \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho} + \equiv \frac{1}{2} |\va{v}|^2 + \Phi + \frac{p}{\rho} } \end{aligned}$$ Where $$\Phi$$ is the gravitational potential, such that $$\va{g} = - \nabla \Phi$$. -To prove this theorem, we take the material derivative of $$H$$: +To prove this theorem, we take the +[material derivative](/know/concept/material-derivative/) of $$H$$: $$\begin{aligned} \frac{\mathrm{D} H}{\mathrm{D} t} @@ -50,7 +35,7 @@ $$\begin{aligned} + \frac{1}{\rho} \frac{\mathrm{D} p}{\mathrm{D} t} \end{aligned}$$ -In the first term we insert the Euler equation, +In the first term we insert the [Euler equation](/know/concept/euler-equations/), and in the other two we expand the derivatives: $$\begin{aligned} diff --git a/source/know/concept/central-limit-theorem/index.md b/source/know/concept/central-limit-theorem/index.md index 42bc05b..0ebad36 100644 --- a/source/know/concept/central-limit-theorem/index.md +++ b/source/know/concept/central-limit-theorem/index.md @@ -17,7 +17,8 @@ and calculating $$M$$ averages $$\mu_m$$ (which involves summing over $$N$$), the resulting means $$\mu_m$$ are normally distributed across the $$M$$ samples if $$N$$ is sufficiently large. -More formally, for $$N$$ independent variables $$x_n$$ with probability distributions $$p(x_n)$$, +More formally, for $$N$$ independent variables $$x_n$$ +with probability distributions $$p(x_n)$$, we define the following totals of all variables, means and variances: $$\begin{aligned} @@ -39,9 +40,9 @@ $$\begin{aligned} } \end{aligned}$$ -We prove this below, -but first we need to introduce some tools. -Given a probability density $$p(x)$$, its [Fourier transform](/know/concept/fourier-transform/) +We prove this below, but first we need to introduce some tools. +Given a probability density $$p(x)$$, +its [Fourier transform](/know/concept/fourier-transform/) is called the **characteristic function** $$\phi(k)$$: $$\begin{aligned} @@ -70,7 +71,8 @@ $$\begin{aligned} = i^n \: \overline{x^n} \end{aligned}$$ -Next, the **cumulants** $$C^{(n)}$$ are defined from the Taylor expansion of $$\ln\!\big(\phi(k)\big)$$: +Next, the **cumulants** $$C^{(n)}$$ are defined +from the Taylor expansion of $$\ln\!\big(\phi(k)\big)$$: $$\begin{aligned} \ln\!\big( \phi(k) \big) @@ -96,9 +98,8 @@ $$\begin{aligned} = - \overline{x}^2 + \overline{x^2} = \sigma^2 \end{aligned}$$ -Now that we have introduced these tools, -we define $$t$$ as the sum -of $$N$$ independent variables $$x_n$$, in other words: +Now that we have introduced these tools, we repeat our definition of $$t$$ +as the sum of $$N$$ independent variables $$x_n$$, in other words: $$\begin{aligned} t @@ -116,10 +117,11 @@ $$\begin{aligned} &= \Big( p_1 * \big( p_2 * ( ... * (p_N * \delta))\big)\Big)(t) \end{aligned}$$ -In other words, the integrals pick out all combinations of $$x_n$$ which -add up to the desired $$t$$-value, and multiply the probabilities -$$p(x_1) p(x_2) \cdots p(x_N)$$ of each such case. This is a convolution, -so the [convolution theorem](/know/concept/convolution-theorem/) +In other words, we integrate over all possible combinations of $$x_n$$, +and use the Dirac delta function to pick out the combinations +where the $$x_n$$ add up to the desired $$t$$-value, +and multiply the probabilities $$p(x_1) \, p(x_2) \cdots p(x_N)$$ of each such case. +This is a convolution, so the [convolution theorem](/know/concept/convolution-theorem/) states that it is a product in the Fourier domain: $$\begin{aligned} diff --git a/source/know/concept/clausius-mossotti-relation/index.md b/source/know/concept/clausius-mossotti-relation/index.md index 03bdcac..61332db 100644 --- a/source/know/concept/clausius-mossotti-relation/index.md +++ b/source/know/concept/clausius-mossotti-relation/index.md @@ -18,19 +18,21 @@ $$\begin{aligned} \end{aligned}$$ If there are $$N$$ such bodies per unit volume, -the polarization density $$\vb{P} = \varepsilon_0 \chi_e \vb{E}$$ -with $$\vb{P} = N \vb{p}$$ suggests that $$\chi_e = N \alpha$$. -However, this is an underestimation: +the macroscopic polarization density $$\vb{P} = \varepsilon_0 \chi_e \vb{E}$$ +with $$\vb{P} = N \vb{p}$$ may suggest that $$\chi_e = N \alpha$$. +However, this turns out to be an underestimation: each body's induced dipole creates its own electric field, weakening the field felt by its neighbors. -We need to include this somehow, -but $$\alpha$$ is defined for a single dipole in a vacuum. +To calculate $$\chi_e$$ from $$\alpha$$, we need to include this effect, +but $$\alpha$$ is defined only for a single dipole in a vacuum. -Let $$\vb{E}_\mathrm{int}$$ be the uniform internal field excluding the dipoles' contributions, -and $$\vb{E}(\vb{r})$$ the net field including them. +Let $$\vb{E}_\mathrm{int}$$ be the uniform internal field +excluding the dipoles' contributions, +and $$\vb{E}(\vb{r})$$ be the net field including them. Assume that the dipoles $$\vb{p}_i$$ are arranged in a regular crystal lattice at sites $$\vb{R}_i$$. -Then $$\vb{E}(\vb{r})$$ is the sum of $$\vb{E}_\mathrm{int}$$ and all the dipoles' fields: +Then $$\vb{E}(\vb{r})$$ is the sum of $$\vb{E}_\mathrm{int}$$ +and all the dipoles' fields: $$\begin{aligned} \vb{E}(\vb{r}) @@ -45,6 +47,7 @@ $$\begin{aligned} = - \frac{1}{4 \pi \varepsilon_0} \nabla \bigg( \frac{\vu{r} \cdot \vb{p}_i}{|\vb{r}|^2} \bigg) \end{aligned}$$ + {% include proof/start.html id="proof-dipole" -%} The atoms or molecules $$\vb{p}_i$$ need not be perfect dipoles, as long as they approximate one when viewed from a distance @@ -84,6 +87,7 @@ $$\begin{aligned} Then the corresponding electric field $$\vb{E}_i$$ is given by $$- \nabla V_i$$ as is well known. {% include proof/end.html id="proof-dipole" -%} + The dipole $$\vb{p}_0$$ at $$\vb{r} = 0$$ feels a net local field $$\vb{E}_\mathrm{loc}$$, given below. The crystal's symmetry ensures that all its neighbors' fields cancel out: @@ -101,9 +105,10 @@ $$\begin{aligned} Even if there is no regular lattice, this result still holds well enough, as long as the dipoles are uniformly distributed over a large volume. -So what was the point of including $$\vb{E}_i(\vb{r})$$ in the first place? -Well, keep in mind that the sum over neighbors is nonzero for $$\vb{r} \neq \vb{R}_i$$, -which *does* affect the macroscopic field $$\vb{E}$$, defined as: +So... if all the neighbors' fields cancel out at $$\vb{r} \in \vb{R}_i$$, +then what was the point of including $$\vb{E}_i(\vb{r})$$ in the first place? +Well, those contributions do *not* cancel out for $$\vb{r} \not{\!\!\in} \: \vb{R}_i$$, +and this fact *does* affect the average macroscopic field $$\vb{E}$$, defined as: $$\begin{aligned} \vb{E} diff --git a/source/know/concept/ehrenfests-theorem/index.md b/source/know/concept/ehrenfests-theorem/index.md index fba0192..14d17fa 100644 --- a/source/know/concept/ehrenfests-theorem/index.md +++ b/source/know/concept/ehrenfests-theorem/index.md @@ -8,20 +8,19 @@ categories: layout: "concept" --- -In quantum mechanics, **Ehrenfest's theorem** gives a general expression for the -time evolution of an observable's expectation value $$\expval{\hat{L}}$$. - -The time-dependent Schrödinger equation is as follows, +In quantum mechanics, **Ehrenfest's theorem** gives a general expression +for the time evolution of an observable's expectation value $$\expval{\hat{L}}$$. +Recall the time-dependent Schrödinger equation, where prime denotes differentiation with respect to time $$t$$: $$\begin{aligned} \Ket{\psi'} = \frac{1}{i \hbar} \hat{H} \Ket{\psi} - \qquad + \qquad \qquad \Bra{\psi'} = - \frac{1}{i \hbar} \Bra{\psi} \hat{H} \end{aligned}$$ Given an observable operator $$\hat{L}$$ and a state $$\Ket{\psi}$$, -the time-derivative of the expectation value $$\expval{\hat{L}}$$ is as follows +the $$t$$-derivative of the expectation value $$\expval{\hat{L}}$$ is as follows (due to the product rule of differentiation): $$\begin{aligned} @@ -43,28 +42,26 @@ $$\begin{aligned} } \end{aligned}$$ -In practice, since most operators are time-independent, -the last term often vanishes. - -As a interesting side note, in the [Heisenberg picture](/know/concept/heisenberg-picture/), -this relation proves itself, -when one simply wraps all terms in $$\Bra{\psi}$$ and $$\Ket{\psi}$$. +In practice, since most operators are time-independent, the last term often vanishes. +Note that this relation is trivial to prove +in the [Heisenberg picture](/know/concept/heisenberg-picture/), +by wrapping all terms in $$\Bra{\psi}$$ and $$\Ket{\psi}$$. -Two observables of particular interest are the position $$\hat{X}$$ and momentum $$\hat{P}$$. -Applying the above theorem to $$\hat{X}$$ yields the following, -which we reduce using the fact that $$\hat{X}$$ commutes -with the potential $$V(\hat{X})$$, -because one is a function of the other: +Two observables of particular interest +are position $$\hat{X}$$ and momentum $$\hat{P}$$. +Applying the theorem to $$\hat{X}$$ yields the following, +using $$\hat{H} = \hat{P}^2 / (2 m) + V(\hat{X})$$ +and a few basic properties of commutators: $$\begin{aligned} \dv{\expval{\hat{X}}}{t} &= \frac{1}{i \hbar} \Expval{[\hat{X}, \hat{H}]} - = \frac{1}{2 i \hbar m} \Expval{[\hat{X}, \hat{P}^2] + 2 m [\hat{X}, V(\hat{X})]} - = \frac{1}{2 i \hbar m} \Expval{[\hat{X}, \hat{P}^2]} + \\ + &= \frac{1}{2 i \hbar m} \Expval{[\hat{X}, \hat{P}^2] + 2 m [\hat{X}, V(\hat{X})]} \\ &= \frac{1}{2 i \hbar m} \Expval{\hat{P} [\hat{X}, \hat{P}] + [\hat{X}, \hat{P}] \hat{P}} - = \frac{2 i \hbar}{2 i \hbar m} \expval{\hat{P}} - = \frac{\expval{\hat{P}}}{m} + \\ + &= \frac{2 i \hbar}{2 i \hbar m} \expval{\hat{P}} \end{aligned}$$ This is the first part of the "original" form of Ehrenfest's theorem, @@ -72,7 +69,8 @@ which is reminiscent of classical Newtonian mechanics: $$\begin{gathered} \boxed{ - \dv{\expval{\hat{X}}}{t} = \frac{\expval{\hat{P}}}{m} + \dv{\expval{\hat{X}}}{t} + = \frac{\expval{\hat{P}}}{m} } \end{gathered}$$ @@ -82,33 +80,32 @@ gives us: $$\begin{aligned} \dv{\expval{\hat{P}}}{t} &= \frac{1}{i \hbar} \Expval{[\hat{P}, \hat{H}]} - = \frac{1}{2 i \hbar m} \Expval{[\hat{P}, \hat{P}^2] + 2 m [\hat{P}, V(\hat{X})]} - = \frac{1}{i \hbar} \Expval{[\hat{P}, V(\hat{X})]} + \\ + &= \frac{1}{2 i \hbar m} \Expval{[\hat{P}, \hat{P}^2] + 2 m [\hat{P}, V(\hat{X})]} + \\ + &= \frac{1}{i \hbar} \Expval{[\hat{P}, V(\hat{X})]} \end{aligned}$$ -To find the commutator, we go to the $$\hat{X}$$-basis and use a test -function $$f(x)$$: +To evaluate the commutator, +we go to the $$\hat{X}$$-basis and use a test function $$f(x)$$: $$\begin{aligned} \Comm{- i \hbar \dv{}{x}}{V(x)} \: f(x) + &= - i \hbar \dv{}{x} \Big( V(x) \: f(x) \Big) - V(x) \Big( \!-\! i \hbar \dv{}{x} \Big) f(x) + \\ &= - i \hbar \frac{dV}{dx} f(x) - i \hbar V(x) \frac{df}{dx} + i \hbar V(x) \frac{df}{dx} - = - i \hbar \frac{dV}{dx} f(x) -\end{aligned}$$ - -By inserting this result back into the previous equation, we find the following: - -$$\begin{aligned} - \dv{\expval{\hat{P}}}{t} - &= - \frac{i \hbar}{i \hbar} \Expval{\frac{d V}{d \hat{X}}} - = - \Expval{\frac{d V}{d \hat{X}}} + \\ + &= - i \hbar \frac{dV}{dx} f(x) \end{aligned}$$ -This is the second part of Ehrenfest's theorem, -which is also similar to Newtonian mechanics: +By inserting this result back into the previous equation, +we find the second part of Ehrenfest's original theorem, +which is again reminiscent Newtonian mechanics: $$\begin{gathered} \boxed{ - \dv{\expval{\hat{P}}}{t} = - \Expval{\pdv{V}{\hat{X}}} + \dv{\expval{\hat{P}}}{t} + = - \Expval{\pdv{V}{\hat{X}}} } \end{gathered}$$ @@ -121,7 +118,7 @@ $$\begin{gathered} \Expval{\pdv{\hat{H}}{\hat{P}}} = \dv{\expval{\hat{X}}}{t} } - \qquad \quad + \qquad \qquad \boxed{ - \Expval{\pdv{\hat{H}}{\hat{X}}} = \dv{\expval{\hat{P}}}{t} diff --git a/source/know/concept/ficks-laws/index.md b/source/know/concept/ficks-laws/index.md index 8d5da7d..20bc50b 100644 --- a/source/know/concept/ficks-laws/index.md +++ b/source/know/concept/ficks-laws/index.md @@ -21,10 +21,12 @@ as opposed to **non-Fickian** or **anomalous diffusion**. moves from regions of high concentration to regions of lower concentration, at a rate proportional to the difference in concentration. -Let $$\vec{J}$$ be the **diffusion flux** (with unit $$\mathrm{m}^{-2} \mathrm{s}^{-1}$$), +Let $$\vec{J}$$ be the **diffusion flux** +(with unit $$\mathrm{m}^{-2} \mathrm{s}^{-1}$$), whose magnitude and direction describes the "flow" of diffusing matter. Formally, Fick's first law predicts that the flux -is proportional to the gradient of the concentration $$C$$ (with unit $$\mathrm{m}^{-3}$$): +is proportional to the gradient of the concentration $$C(\vec{r})$$ +(with unit $$\mathrm{m}^{-3}$$): $$\begin{aligned} \boxed{ @@ -37,12 +39,9 @@ Where $$D$$ (with unit $$\mathrm{m}^{2}/\mathrm{s}$$) is known as the **diffusion coefficient** or **diffusivity**, and depends on both the medium and the diffusing substance. -Fick's first law is a general physical principle, -which was discovered experimentally, -and thus does not have a general derivation. -Proofs for specific systems do exist, -but they say more about those systems -than about diffusion in general. +Fick's first law is an empirical physical principle, +and therefore does not have a general derivation, +although proofs for specific systems do exist. @@ -59,12 +58,12 @@ $$\begin{aligned} \end{aligned}$$ Over time $$t$$, matter enters/leaves $$V$$. -Let $$S$$ be the surface of $$V$$, and $$\vec{J}$$ the diffusion flux, -then $$M$$ changes as follows, to which we apply the divergence theorem: +Let $$\partial V$$ be the surface of $$V$$, and $$\vec{J}$$ the diffusion flux, +then $$M$$ changes as follows, applying the divergence theorem: $$\begin{aligned} \dv{M}{t} - = - \int_S \vec{J} \cdot \dd{\vec{S}} + = - \int_{\partial V} \vec{J} \cdot \dd{\vec{S}} = - \int_V \nabla \cdot \vec{J} \dd{V} \end{aligned}$$ @@ -91,7 +90,7 @@ the general form of Fick's second law: $$\begin{aligned} \boxed{ \pdv{C}{t} - = \nabla \cdot \Big( D \: \nabla C \Big) + = \nabla \cdot \Big( D \, \nabla C \Big) } \end{aligned}$$ @@ -100,7 +99,8 @@ with respect to space $$\vec{r}$$ and concentration $$C$$, in which case Fick's second law reduces to: $$\begin{aligned} - \pdv{C}{t} = D \: \nabla^2 C + \pdv{C}{t} + = D \, \nabla^2 C \end{aligned}$$ @@ -108,7 +108,7 @@ $$\begin{aligned} ## Fundamental solution Fick's second law has exact solutions for many situations, -but the most important one is arguably the **fundamental solution**. +but the most important one is arguably the **fundamental solution** $$H$$. Consider a 1D system (for simplicity) with constant diffusivity $$D$$, where the initial concentration $$C(x, 0)$$ is a [Dirac delta function](/know/concept/dirac-delta-function/): @@ -118,8 +118,8 @@ $$\begin{aligned} = \delta(x - x_0) \end{aligned}$$ -By solving Fick's second law with this initial condition, -$$C$$'s time evolution turns out to be: +By solving Fick's second law with this initial condition (details omitted), +we find that $$C$$ obeys: $$\begin{aligned} H(x - x_0, t) @@ -127,13 +127,12 @@ $$\begin{aligned} = \frac{1}{\sqrt{4 \pi D t}} \exp\!\Big( \!-\!\frac{(x - x_0)^2}{4 D t} \Big) \end{aligned}$$ -This result is a normalized Gaussian, -as a consequence of -the [central limit theorem](/know/concept/central-limit-theorem/): -the diffusion behaviour is a sum of many independent steps -(i.e. molecular collisions). +This result is a normalized Gaussian: +diffusion is a sum of many independent molecular collisions, +so the [central limit theorem](/know/concept/central-limit-theorem/) applies, +hence this result. The standard deviation is $$\sqrt{2 D t}$$, -meaning that the distance of a diffusion is proportional to $$\sqrt{t}$$. +meaning that the expected distance of a diffusion is proportional to $$\sqrt{t}$$. This solution $$H$$ is extremely useful, because any initial concentration $$C(x, 0)$$ can be written as @@ -146,22 +145,19 @@ $$\begin{aligned} \end{aligned}$$ In other words, any function is a linear combination of delta functions. -Fick's second law is linear, -so the overall solution $$C(x, t)$$ is the same combination of fundamental solutions $$H$$: +Fick's second law is linear, so the overall solution $$C(x, t)$$ +is the same combination of fundamental solutions $$H$$: $$\begin{aligned} C(x, t) = (C * H)(x) &= \int_{-\infty}^\infty C(x_0, 0) \: H(x - x_0, t) \dd{x_0} - \\ - &= \int_{-\infty}^\infty \frac{1}{\sqrt{4 \pi D t}} \exp\!\Big( \!-\!\frac{(x - x_0)^2}{4 D t} \Big) \: C(x_0, 0) \dd{x_0} \end{aligned}$$ This technique is analogous to using the [impulse response](/know/concept/impulse-response/) -of a linear operator to extrapolate all its inhomogeneous solutions. -The difference is that here, we used the initial condition -instead of the forcing function. +of a linear operator to extrapolate all its inhomogeneous solutions, +but here we used the initial condition instead of the forcing function. diff --git a/source/know/concept/grad-shafranov-equation/index.md b/source/know/concept/grad-shafranov-equation/index.md index b86c032..c9104d2 100644 --- a/source/know/concept/grad-shafranov-equation/index.md +++ b/source/know/concept/grad-shafranov-equation/index.md @@ -36,9 +36,9 @@ $$\begin{aligned} = 0 \end{aligned}$$ -Notice that $$\vb{E} = 0$$ is a result of the ideal generalized Ohm's law. -Under these assumptions, the relevant MHD equations to be solved are -Gauss' law for magnetism, Ampère's law, and the MHD momentum equation, respectively: +Notice that $$\vb{E} = 0$$ is a result of ideal MHD's generalized Ohm's law. +Under these assumptions, the relevant equations to be solved are +Gauss' law for magnetism, Ampère's law, and the momentum equation of MHD, respectively: $$\begin{aligned} 0 @@ -51,11 +51,12 @@ $$\begin{aligned} = \vb{J} \cross \vb{B} \end{aligned}$$ -The goal is to analyze them in this order, +The idea is to analyze them in this order, exploiting toroidal symmetry along the way, to arrive at a general equilibrium condition. -[Cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/) $$(r, \theta, z)$$ -are a natural choice, with the $$z$$-axis running through the middle of the torus. +[Cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/) +$$(r, \theta, z)$$ are a natural choice, +with the $$z$$-axis running through the middle of the torus. As preparation, it is a good idea to write $$\vb{B}$$ as the curl of a magnetic vector potential $$\vb{A}$$, diff --git a/source/know/concept/holomorphic-function/index.md b/source/know/concept/holomorphic-function/index.md index 976758b..db3bdfb 100644 --- a/source/know/concept/holomorphic-function/index.md +++ b/source/know/concept/holomorphic-function/index.md @@ -28,7 +28,8 @@ $$\begin{aligned} } \end{aligned}$$ -We decompose $$f$$ into the real functions $$u$$ and $$v$$ of real variables $$x$$ and $$y$$: +We decompose $$f$$ into the real functions $$u$$ and $$v$$ +of real variables $$x$$ and $$y$$: $$\begin{aligned} f(z) @@ -36,7 +37,8 @@ $$\begin{aligned} = u(x, y) + i v(x, y) \end{aligned}$$ -Since we are free to choose the direction of $$\Delta z$$, we choose $$\Delta x$$ and $$\Delta y$$: +Since we are free to choose the direction of $$\Delta z$$, +we choose $$\Delta x$$ and $$\Delta y$$: $$\begin{aligned} f'(z) @@ -53,9 +55,13 @@ we thus arrive at the **Cauchy-Riemann equations**: $$\begin{aligned} \boxed{ - \pdv{u}{x} = \pdv{v}{y} - \qquad - \pdv{v}{x} = - \pdv{u}{y} + \begin{aligned} + \pdv{u}{x} + &= \pdv{v}{y} + \\ + \pdv{v}{x} + &= - \pdv{u}{y} + \end{aligned} } \end{aligned}$$ @@ -85,7 +91,8 @@ Just like before, we decompose $$f(z)$$ into its real and imaginary parts: $$\begin{aligned} \oint_C f(z) \dd{z} &= \oint_C (u + i v) \dd{(x + i y)} - = \oint_C (u + i v) \:(\dd{x} + i \dd{y}) + \\ + &= \oint_C (u + i v) \:(\dd{x} + i \dd{y}) \\ &= \oint_C u \dd{x} - v \dd{y} + i \oint_C v \dd{x} + u \dd{y} \end{aligned}$$ @@ -137,10 +144,10 @@ $$\begin{aligned} {% include proof/end.html id="proof-int-formula" %} -Similarly, **Cauchy's differentiation formula**, -or **Cauchy's integral formula for derivatives** -gives all derivatives of a holomorphic function as follows, -and also guarantees their existence: +Similarly, **Cauchy's differentiation formula** +or **integral formula for derivatives** +gives the $$n$$th-order derivative of a holomorphic function as follows, +and guarantees its existence: $$\begin{aligned} \boxed{ diff --git a/source/know/concept/larmor-precession/index.md b/source/know/concept/larmor-precession/index.md index 601dae7..e29432b 100644 --- a/source/know/concept/larmor-precession/index.md +++ b/source/know/concept/larmor-precession/index.md @@ -11,28 +11,29 @@ layout: "concept" Consider a stationary spin-1/2 particle, placed in a [magnetic field](/know/concept/magnetic-field/) with magnitude $$B$$ pointing in the $$z$$-direction. -In that case, its Hamiltonian $$\hat{H}$$ is given by: +In that case, the Hamiltonian $$\hat{H}$$ is given by: $$\begin{aligned} \hat{H} = - \gamma B \hat{S}_z = - \frac{\hbar}{2} \gamma B \hat{\sigma_z} \end{aligned}$$ -Where $$\gamma = - q / m$$ is the gyromagnetic ratio, +Where $$\gamma = q g / (2 m)$$ is the so-called *gyromagnetic ratio* +for a particle with charge $$q$$, mass $$m$$, +and a system-dependent *$$g$$-factor* (for electrons $$g \approx 2$$), and $$\hat{\sigma}_z$$ is the Pauli spin matrix for the $$z$$-direction. -Since $$\hat{H}$$ is proportional to $$\hat{\sigma}_z$$, -they share eigenstates $$\Ket{\downarrow}$$ and $$\Ket{\uparrow}$$. -The respective eigenenergies $$E_{\downarrow}$$ and $$E_{\uparrow}$$ are as follows: +Because $$\hat{H}$$ is proportional to $$\hat{\sigma}_z$$, +they share eigenstates $$\Ket{\downarrow}$$ and $$\Ket{\uparrow}$$, +so the respective eigenenergies $$E_{\downarrow}$$ and $$E_{\uparrow}$$ are as follows: $$\begin{aligned} E_{\downarrow} = \frac{\hbar}{2} \gamma B - \qquad + \qquad \qquad E_{\uparrow} = - \frac{\hbar}{2} \gamma B \end{aligned}$$ Because $$\hat{H}$$ is time-independent, the general time-dependent solution $$\Ket{\chi(t)}$$ is of the following form, -where $$a$$ and $$b$$ are constants, -and the exponentials are "twiddle factors": +where $$a$$ and $$b$$ are constants: $$\begin{aligned} \Ket{\chi(t)} @@ -72,7 +73,7 @@ $$\begin{aligned} \\ &= \frac{\hbar}{2} \cos(\theta/2) \sin(\theta/2) \Big( e^{i \gamma B t} + e^{- i \gamma B t} \Big) \\ - &= \frac{\hbar}{2} \cos(\gamma B t) \cdot 2 \cos(\theta/2) \sin(\theta/2) + &= \frac{\hbar}{2} \cos(\theta/2) \sin(\theta/2) \cdot 2 \cos(\gamma B t) \\ &= \frac{\hbar}{2} \sin(\theta) \cos(\gamma B t) \end{aligned}$$ @@ -82,12 +83,13 @@ with the following results: $$\begin{aligned} \matrixel{\chi}{\hat{S}_y}{\chi} = - \frac{\hbar}{2} \sin(\theta) \sin(\gamma B t) - \qquad + \qquad \qquad \matrixel{\chi}{\hat{S}_z}{\chi} = \frac{\hbar}{2} \cos(\theta) \end{aligned}$$ -The result is that the spin axis is off by $$\theta$$ from the $$z$$-direction, -and is rotating (or **precessing**) around the $$z$$-axis at the **Larmor frequency** $$\omega$$: +The result is that, if the spin axis is off by $$\theta$$ from the $$z$$-direction, +then it rotates (or **precesses**) around the $$z$$-axis +at the **Larmor frequency** $$\omega$$: $$\begin{aligned} \boxed{ diff --git a/source/know/concept/lubrication-theory/index.md b/source/know/concept/lubrication-theory/index.md index 4015526..54200d3 100644 --- a/source/know/concept/lubrication-theory/index.md +++ b/source/know/concept/lubrication-theory/index.md @@ -9,8 +9,7 @@ categories: layout: "concept" --- -**Lubricants** are widely used -to reduce friction between two moving surfaces. +**Lubricants** are widely used to reduce friction between two moving surfaces. In fluid mechanics, **lubrication theory** is the study of fluids that are tightly constrained in one dimension, especially those in small gaps between moving surfaces. @@ -34,32 +33,32 @@ $$\begin{aligned} \approx \frac{d^2}{L^2} \mathrm{Re} \end{aligned}$$ -If $$d$$ is small enough compared to $$L$$, -then $$\mathrm{Re}_\mathrm{gap} \ll 1$$. +If $$d$$ is small enough compared to $$L$$, then $$\mathrm{Re}_\mathrm{gap} \ll 1$$. More formally, we need $$d \ll L / \sqrt{\mathrm{Re}}$$, so we are inside the boundary layer, in the realm of the [Prandtl equations](/know/concept/prandtl-equations/). Let $$\mathrm{Re}_\mathrm{gap} \ll 1$$. -We are thus dealing with *Stokes flow*, in which case +We are then dealing with *Stokes flow*, in which case the [Navier-Stokes equations](/know/concept/navier-stokes/equations/) can be reduced to the following *Stokes equations*: $$\begin{aligned} \pdv{p}{x} = \eta \: \Big( \pdvn{2}{v_x}{x} + \pdvn{2}{v_x}{y} \Big) - \qquad \quad + \qquad \qquad \pdv{p}{y} = \eta \: \Big( \pdvn{2}{v_y}{x} + \pdvn{2}{v_y}{y} \Big) \end{aligned}$$ -Let the $$y = 0$$ plane be an infinite flat surface, +Let the $$y = 0$$ plane be an infinite flat surface +(a good approximation because $$d \ll L$$), sliding in the positive $$x$$-direction at a constant velocity $$U$$. -On the other side of the gap, -an arbitrary surface is described by $$h(x)$$. +On the other side of the gap, an arbitrary surface +is described by a height function $$h(x)$$. Since the gap is so narrow, -and the surfaces' movements cause large shear stresses inside, +and the surfaces' movements cause large shear stresses inside it, $$v_y$$ is negligible compared to $$v_x$$. Furthermore, because the gap is so long, we assume that $$\ipdv{v_x}{x}$$ is negligible compared to $$\ipdv{v_x}{y}$$. @@ -68,7 +67,7 @@ This reduces the Stokes equations to: $$\begin{aligned} \pdv{p}{x} = \eta \pdvn{2}{v_x}{y} - \qquad \quad + \qquad \qquad \pdv{p}{y} = 0 \end{aligned}$$ @@ -92,18 +91,20 @@ $$\begin{aligned} \end{aligned}$$ The moving bottom surface drags fluid in the $$x$$-direction -at a volumetric rate $$Q$$, given by: +at a volumetric rate $$Q(x)$$, given by: $$\begin{aligned} - Q - = \int_0^{h(x)} v_x(x, y) \dd{y} - = \bigg[ \frac{p'}{6 \eta} y^3 - \frac{p'}{4 \eta} h y^2 - \frac{U}{2 h} y^2 + U y \bigg]_0^{h} - = - \frac{p'}{12 \eta} h^3 + \frac{U}{2} h + Q(x) + &= \int_0^{h(x)} v_x(x, y) \dd{y} + \\ + &= \bigg[ \frac{p'}{6 \eta} y^3 - \frac{p'}{4 \eta} h y^2 - \frac{U}{2 h} y^2 + U y \bigg]_0^{h} + \\ + &= - \frac{p'}{12 \eta} h^3(x) + \frac{U}{2} h(x) \end{aligned}$$ -Assuming that the lubricant is incompressible, -meaning that the same volume of fluid must be leaving a point as is entering it. -In other words, $$Q$$ is independent of $$x$$, +Let us assume that the lubricant is incompressible, +meaning that the same volume of fluid must be both leaving and entering the gap. +In that case, $$Q$$ is independent of $$x$$, which allows us to write $$p'(x)$$ in terms of measurable constants and the known function $$h(x)$$: @@ -139,7 +140,8 @@ $$\begin{aligned} = - 2 h' \frac{U h - 3 Q}{h^4} \big( 2 h y - 3 y^2 \big) \end{aligned}$$ -Integrating with respect to $$y$$ thus leads to the following transverse velocity $$v_y$$: +Integrating with respect to $$y$$ therefore leads to +the following transverse velocity $$v_y$$: $$\begin{aligned} \boxed{ @@ -148,10 +150,11 @@ $$\begin{aligned} } \end{aligned}$$ -Typically, the lubricant is not in a preexisting pressure differential, -i.e it is not getting pumped through the system. -Although the pressure gradient $$p'$$ need not be zero, -we therefore expect that its integral vanishes: +Usually, the lubricant is not getting pumped through the system. +In that case, although the pressure gradient $$p'$$ need not be zero in all points +(i.e. there may be complex dynamics inside the gap), +we do expect that its integral across the gap vanishes +(because both sides are at the same pressure): $$\begin{aligned} 0 @@ -164,7 +167,7 @@ Isolating this for $$Q$$, and defining $$q$$ as below, yields a simple equation: $$\begin{aligned} Q = \frac{1}{2} U q - \qquad \quad + \qquad \qquad q \equiv \frac{\int_L h^{-2} \dd{x}}{\int_L h^{-3} \dd{x}} \end{aligned}$$ @@ -178,27 +181,25 @@ $$\begin{aligned} &= U \Big( 1 - \frac{y}{h} \Big) \Big( 1 - \frac{3 y (h - q)}{h^2} \Big) \end{aligned}$$ -The first factor is always positive, -but the second can be negative, -if for some $$y$$-values: +The first factors are always positive, but the last one can be negative, +if any $$y$$-values satisfy: $$\begin{aligned} h^2 < 3 y (h - q) - \quad \implies \quad + \qquad \implies \qquad y > \frac{h^2}{3 (h - q)} \end{aligned}$$ -Since $$h > y$$, such $$y$$-values will only exist +Since $$h \le y$$, such $$y$$-values will only exist if $$h$$ is larger than some threshold: $$\begin{aligned} 3 (h - q) > h - \quad \implies \quad + \qquad \implies \qquad h > \frac{3}{2} q \end{aligned}$$ -If this condition is satisfied, -there will be some flow reversal: +If this condition is satisfied, there will be some flow reversal: rather than just getting dragged by the shearing motion, the lubricant instead "rolls" inside the gap. This is confirmed by $$v_y$$: diff --git a/source/know/concept/lyddane-sachs-teller-relation/index.md b/source/know/concept/lyddane-sachs-teller-relation/index.md index 9cec9dc..60c8984 100644 --- a/source/know/concept/lyddane-sachs-teller-relation/index.md +++ b/source/know/concept/lyddane-sachs-teller-relation/index.md @@ -20,7 +20,7 @@ creating lattice vibrations (phonons), i.e. a photon-phonon conversion, where the total energy and momentum must be conserved. If the photon has frequency $$\omega$$ and wavenumber $$k$$, -and the phonon $$\Omega$$ and $$K$$, then: +and the phonon has $$\Omega$$ and $$K$$, then: $$\begin{aligned} \hbar \omega @@ -71,8 +71,8 @@ $$\begin{aligned} Where $$\vb{E}(t) = \vb{E}_0 e^{- i \omega t}$$ represents the light, and $$\kappa$$ is the spring constant of the polar bonds' restoring force. -Note that the latter depends on the displacement between the ions, -instead of from their equilibrium position, +The latter depends on the displacement between the ions, +instead of their displacement from their equilibrium position, so we need to write $$\vb{x}_{+} - \vb{x}_{-}$$ instead of $$\vb{x}_{+}$$. Respectively dividing the equations by $$m_{+}$$ and $$m_{-}$$ @@ -95,10 +95,10 @@ $$\begin{aligned} \end{aligned}$$ Note that $$\Omega_\mathrm{TO}$$ is the phonon frequency for $$K = 0$$. -This is because IR light waves are much larger than the crystal's unit cell, -so we are ignoring all spatial variation in $$\vb{E}$$ -(i.e. the [electric dipole approximation](/know/concept/electric-dipole-approximation/)). -This is equivalent to assuming that $$K \approx 0$$. +This is close enough, because IR light waves are much larger +than the crystal's unit cell, so we can ignore all spatial variation in $$\vb{E}$$ +(the [electric dipole approximation](/know/concept/electric-dipole-approximation/)), +which is equivalent to assuming that $$K = 0$$. For the sake of generality, we also introduce an empirical damping rate $$\gamma$$, @@ -147,7 +147,9 @@ $$\begin{aligned} \end{aligned}$$ In the limits of low and high frequencies $$\omega$$, -we see that $$\varepsilon_r$$ is higher in the former: +we see that $$\varepsilon_r$$ is higher in the former +(also recall that $$\chi_\mathrm{low} > \chi_\mathrm{high}$$ +according to the original Lorentz oscillator model): $$\begin{aligned} \varepsilon_{\mathrm{low}} @@ -159,7 +161,8 @@ $$\begin{aligned} = 1 + \chi_\mathrm{high} \end{aligned}$$ -We can use these quantities to rewrite the relative permittivity $$\varepsilon_r$$ as follows: +We can use these quantities to rewrite +the relative permittivity $$\varepsilon_r$$ as follows: $$\begin{aligned} \varepsilon_r(\omega) @@ -167,9 +170,8 @@ $$\begin{aligned} \frac{\Omega_\mathrm{TO}^2}{\Omega_\mathrm{TO}^2 - \omega^2 - i \gamma \omega} \end{aligned}$$ -For weak damping $$\gamma \approx 0$$, there exists a frequency, -which we will call $$\Omega_\mathrm{LO}$$ in anticipation, -where the dielectric function is zero: +For weak damping $$\gamma \approx 0$$, there exists a frequency +with zero permittivity, which we call $$\Omega_\mathrm{LO}$$: $$\begin{aligned} 0 @@ -179,8 +181,8 @@ $$\begin{aligned} \end{aligned}$$ The physical significance of $$\varepsilon_r = 0$$ can be -seen from [Gauss' law](/know/concept/maxwells-equations), under the assumption that there is -no net charge density: +seen from [Gauss' law](/know/concept/maxwells-equations), +under the assumption that there is no net charge density: $$\begin{aligned} \nabla \cdot \vb{D} @@ -188,10 +190,10 @@ $$\begin{aligned} = 0 \end{aligned}$$ -If $$\varepsilon_r \neq 0$$, then $$\nabla \cdot \vec{E} = 0$$, +If $$\varepsilon_r \neq 0$$, then $$\nabla \cdot \vb{E} = 0$$, corresponding to a transverse light wave as usual. -However, if $$\varepsilon_r = 0$$, then $$\nabla \cdot \vec{E} \neq 0$$, -representing a longitudinal electric wave, like a plasmon in metal. +However, if $$\varepsilon_r = 0$$, then $$\nabla \cdot \vb{E} \neq 0$$, +representing a longitudinal electric wave, analogous to plasmons in metals. Rearranging the equation for $$\Omega_\mathrm{LO}$$ gives us the **Lyddane-Sachs-Teller (LST) relation**: @@ -230,17 +232,18 @@ In practice, real materials have $$\gamma > 0$$, which reduces $$R$$ somewhat. Because the photons and TO phonons interact so strongly for $$\omega \approx \Omega_\mathrm{TO}$$, they can be treated as a single **phonon polariton** there, -with a dispersion relation given by: +with a self-referential dispersion relation given by: $$\begin{aligned} \omega_\mathrm{pp}(K) - = \frac{c}{\sqrt{\varepsilon_r(\omega_\mathrm{pp})}} K + = \frac{c}{\sqrt{\varepsilon_r(\omega_\mathrm{pp}(K))}} K \end{aligned}$$ Earlier, when treating the photon and phonon separately, we wanted the intersection between $$\omega(k)$$ and $$\Omega(K)$$. -But now, for $$\omega_\mathrm{pp}(K)$$, there is none! This is a good example -of the typical *anti-crossing* behavior of strongly coupled systems. +But now, plotting $$\omega_\mathrm{pp}(K)$$ reveals that there is no intersection! +This is a good example of the typical *anti-crossing* +behavior of strongly coupled quantum systems. diff --git a/source/know/concept/prandtl-equations/index.md b/source/know/concept/prandtl-equations/index.md index d82657c..8f5c4d0 100644 --- a/source/know/concept/prandtl-equations/index.md +++ b/source/know/concept/prandtl-equations/index.md @@ -11,13 +11,12 @@ layout: "concept" In fluid dynamics, the **Prandtl equations** or **boundary layer equations** describe the movement of a [viscous](/know/concept/viscosity/) fluid -with a large [Reynolds number](/know/concept/reynolds-number/) $$\mathrm{Re} \gg 1$$ -close to a solid surface. +with a large [Reynolds number](/know/concept/reynolds-number/) +$$\mathrm{Re} \gg 1$$ close to a solid surface. -Fluids with a large Reynolds number -are often approximated as having zero viscosity, -since the simpler [Euler equations](/know/concept/euler-equations) -can then be used instead of the [Navier-Stokes equations](/know/concept/navier-stokes-equations/). +Fluids with a large Reynolds number are often approximated as having zero viscosity, +since the simpler [Euler equations](/know/concept/euler-equations) can then be used +instead of the [Navier-Stokes equations](/know/concept/navier-stokes-equations/). However, in reality, a viscous fluid obeys the *no-slip* boundary condition: at every solid surface the local velocity must be zero. @@ -29,9 +28,9 @@ This is in contrast to the ideal flow far away from the surface. We consider a simple theoretical case in 2D: a large flat surface located at $$y = 0$$ for all $$x \in \mathbb{R}$$, with a fluid *trying* to flow parallel to it at $$U$$. -The 2D treatment can be justified by assuming that everything is constant in the $$z$$-direction. -We will not solve this case, -but instead derive general equations +The 2D treatment can be justified by assuming +that everything is constant in the $$z$$-direction. +We will not solve this case, but instead derive general equations to describe the flow close to a flat surface. At the wall, there is a very thin boundary layer of thickness $$\delta$$, @@ -39,8 +38,7 @@ where the fluid is assumed to be completely stationary $$\va{v} = 0$$. We are mainly interested in the region $$\delta < y \ll L$$, where $$L$$ is the distance at which the fluid becomes practically ideal. This the so-called **slip-flow** region, -in which the fluid is not stationary, -but still viscosity-dominated. +in which the fluid is not stationary, but still viscosity-dominated. In 2D, the steady Navier-Stokes equations are as follows, where the flow $$\va{v} = (v_x, v_y)$$: @@ -66,13 +64,13 @@ Let $$\tilde{x}$$ and $$\tilde{y}$$ be dimenionless variables of order $$1$$: $$\begin{aligned} x = L \tilde{x} - \qquad \quad + \qquad \qquad y = \delta \tilde{x} - \qquad \quad + \qquad \qquad \pdv{}{x} = \frac{1}{L} \pdv{}{\tilde{x}} - \qquad \quad + \qquad \qquad \pdv{}{y} = \frac{1}{\delta} \pdv{}{\tilde{y}} \end{aligned}$$ @@ -85,10 +83,10 @@ by [Bernoulli's theorem](/know/concept/bernoullis-theorem/): $$\begin{aligned} v_x = U \tilde{v}_x - \qquad \quad + \qquad \qquad v_y = \frac{U \delta}{L} \tilde{v}_y - \qquad \quad + \qquad \qquad p = \rho U^2 \tilde{p} \end{aligned}$$ @@ -124,15 +122,15 @@ and that faster velocities $$U$$ give thinner layers. Furthermore, we expect *downstream thickening*: with distance $$x$$, viscous stresses slow down the slip-flow, leading to a gradual increase of $$\delta(x)$$. -Some dimensional analysis thus yields the following estimate: +Some dimensional analysis yields the following estimate: $$\begin{aligned} \delta \approx \sqrt{\frac{\nu x}{U}} - \sim \sqrt{\frac{\nu L}{U}} + \approx \sqrt{\frac{\nu L}{U}} \end{aligned}$$ -We thus insert $$\delta = \sqrt{\nu L / U}$$ into the Navier-Stokes equations, giving us: +So we insert $$\delta = \sqrt{\nu L / U}$$ into the Navier-Stokes equations, giving us: $$\begin{aligned} \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} @@ -163,15 +161,14 @@ so we can drop many terms, leaving us with these redimensionalized equations: $$\begin{aligned} v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} = - \frac{1}{\rho} \pdv{p}{x} + \nu \pdvn{2}{v_x}{y} - \qquad \quad + \qquad \qquad \pdv{p}{y} = 0 \end{aligned}$$ The second one tells us that for a given $$x$$-value, -the pressure is the same at the surface -as in the main flow $$y > L$$, where the fluid is ideal. -In the latter regime, we apply Bernoulli's theorem to rewrite $$p$$, +the pressure is the same at the surface as in the ideal main flow $$y > L$$. +In the latter regime, we can use Bernoulli's theorem to rewrite $$p$$, using the *Bernoulli head* $$H$$ and the mainstream velocity $$U(x)$$: $$\begin{aligned} @@ -185,18 +182,19 @@ we arrive at the Prandtl equations: $$\begin{aligned} \boxed{ - v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} - = U \dv{U}{x} + \nu \pdvn{2}{v_x}{y} - \qquad \quad - \pdv{v_x}{x} + \pdv{v_y}{y} - = 0 + \begin{aligned} + v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} + &= U \dv{U}{x} + \nu \pdvn{2}{v_x}{y} + \\ + \pdv{v_x}{x} + \pdv{v_y}{y} + &= 0 + \end{aligned} } \end{aligned}$$ A notable application of these equations is the [Blasius boundary layer](/know/concept/blasius-boundary-layer/), -where the surface in question -is a semi-infinite plane. +where the surface in question is a semi-infinite plane. diff --git a/source/know/concept/probability-current/index.md b/source/know/concept/probability-current/index.md index bd41dab..81ca586 100644 --- a/source/know/concept/probability-current/index.md +++ b/source/know/concept/probability-current/index.md @@ -10,48 +10,49 @@ layout: "concept" In quantum mechanics, the **probability current** describes the movement of the probability of finding a particle at given point in space. -In other words, it treats the particle as a heterogeneous fluid with density $$|\psi|^2$$. -Now, the probability of finding the particle within a volume $$V$$ is: +Basically, it treats the particle as a heterogeneous fluid with density $$|\psi|^2$$. +Clearly, the probability of finding the particle within a volume $$V$$ is: $$\begin{aligned} - P = \int_{V} | \psi |^2 \ddn{3}{\vb{r}} + P = \int_{V} | \psi |^2 \dd{\vb{r}} \end{aligned}$$ -As the system evolves in time, this probability may change, so we take -its derivative with respect to time $$t$$, and when necessary substitute -in the other side of the Schrödinger equation to get: +As the system evolves in time, this probability may change, +so we take its derivative with respect to time $$t$$, +and substitute in the other side of the Schrödinger equation to get: $$\begin{aligned} \pdv{P}{t} - &= \int_{V} \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \ddn{3}{\vb{r}} - = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \ddn{3}{\vb{r}} + &= \int_{V} \bigg( \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \bigg) \dd{\vb{r}} \\ - &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vb{r}) \psi^* \Big) - - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vb{r}) \psi \Big) \ddn{3}{\vb{r}} + &= \frac{i}{\hbar} \int_{V} \bigg( \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \bigg) \dd{\vb{r}} \\ - &= \frac{i \hbar}{2 m} \int_{V} - \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \ddn{3}{\vb{r}} - = - \int_{V} \nabla \cdot \vb{J} \ddn{3}{\vb{r}} + &= \frac{i}{\hbar} \int_{V} \bigg( \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vb{r}) \psi^* \Big) + - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vb{r}) \psi \Big) \bigg) \dd{\vb{r}} + \\ + &= \frac{i \hbar}{2 m} \int_{V} \bigg( \!-\! \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \bigg) \dd{\vb{r}} + \\ + &= - \int_{V} \nabla \cdot \vb{J} \dd{\vb{r}} \end{aligned}$$ -Where we have defined the probability current $$\vb{J}$$ as follows in -the $$\vb{r}$$-basis: +Where we have defined the probability current $$\vb{J}$$ +as follows in the $$\vb{r}$$-basis: $$\begin{aligned} \vb{J} - = \frac{i \hbar}{2 m} (\psi \nabla \psi^* - \psi^* \nabla \psi) - = \mathrm{Re} \Big\{ \psi \frac{i \hbar}{m} \psi^* \Big\} + &\equiv \frac{i \hbar}{2 m} (\psi \nabla \psi^* - \psi^* \nabla \psi) + = \Real\!\bigg\{ \psi \Big( \frac{i \hbar}{m} \Big) \nabla \psi^* \bigg\} \end{aligned}$$ Let us rewrite this using the momentum operator -$$\vu{p} = -i \hbar \nabla$$ as follows, noting that $$\vu{p} / m$$ is -simply the velocity operator $$\vu{v}$$: +$$\vu{p} = -i \hbar \nabla = - \vu{p}^*$$ +as follows, noting that $$\vu{p} / m$$ is simply the velocity operator $$\vu{v}$$: $$\begin{aligned} \boxed{ \vb{J} - = \frac{1}{2 m} ( \psi^* \vu{p} \psi - \psi \vu{p} \psi^*) - = \mathrm{Re} \Big\{ \psi^* \frac{\vu{p}}{m} \psi \Big\} - = \mathrm{Re} \{ \psi^* \vu{v} \psi \} + = \Real\!\Big\{ \psi^* \frac{\vu{p}}{m} \psi \Big\} + = \Real\{ \psi^* \vu{v} \psi \} } \end{aligned}$$ @@ -60,8 +61,8 @@ equation: $$\begin{aligned} \pdv{P}{t} - = \int_{V} \pdv{|\psi|^2}{t} \ddn{3}{\vb{r}} - = - \int_{V} \nabla \cdot \vb{J} \ddn{3}{\vb{r}} + = \int_{V} \pdv{|\psi|^2}{t} \dd{\vb{r}} + = - \int_{V} \nabla \cdot \vb{J} \dd{\vb{r}} \end{aligned}$$ By removing the integrals, we thus arrive at the **continuity equation** @@ -74,16 +75,17 @@ $$\begin{aligned} } \end{aligned}$$ -This states that the total probability is conserved, and is reminiscent of charge -conservation in electromagnetism. In other words, the probability at a -point can only change by letting it "flow" towards or away from it. Thus -$$\vb{J}$$ represents the flow of probability, which is analogous to the -motion of a particle. +This states that the total probability is conserved, +and is reminiscent of electric charge conservation. +In other words, the probability at a point can only change +by "flowing" towards or away from it. +Thus $$\vb{J}$$ represents the flow of probability as if it were a fluid. -As a bonus, this still holds for a particle in an electromagnetic vector -potential $$\vb{A}$$, thanks to the gauge invariance of the Schrödinger -equation. We can thus extend the definition to a particle with charge -$$q$$ in an SI-unit field, neglecting spin: +As a bonus, the continuity relation still holds +for a particle in an electromagnetic vector potential $$\vb{A}$$, +thanks to the gauge invariance of the Schrödinger equation. +We can thus extend the definition to a particle +with charge $$q$$ in an SI-unit field, neglecting spin: $$\begin{aligned} \boxed{ diff --git a/source/know/concept/rabi-oscillation/index.md b/source/know/concept/rabi-oscillation/index.md index 9497ad4..a2cea23 100644 --- a/source/know/concept/rabi-oscillation/index.md +++ b/source/know/concept/rabi-oscillation/index.md @@ -14,12 +14,12 @@ In quantum mechanics, we know from the [amplitude rate equations](/know/concept/amplitude-rate-equations/) that a time-dependent term $$\hat{H}_1$$ in the Hamiltonian affects the state as follows, -where $$c_n(t)$$ are the coefficients of the linear combination +where $$c_n(t)$$ are coefficients of a linear combination of unperturbed basis states $$\ket{n} e^{-i E_n t / \hbar}$$: $$\begin{aligned} i \hbar \dv{c_m}{t} - = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1}{n} e^{i \omega_{mn} t} + = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1(t)}{n} e^{i \omega_{mn} t} \end{aligned}$$ Where $$\omega_{mn} \equiv (E_m \!-\! E_n) / \hbar$$ @@ -36,7 +36,7 @@ $$\begin{aligned} \end{aligned}$$ Where $$\omega_0 \equiv \omega_{ba}$$ is positive. -We assume that $$\hat{H}_1$$ has odd spatial parity, +It is realistic to assume that $$\hat{H}_1$$ has odd spatial parity, in which case [Laporte's selection rule](/know/concept/selection-rules/) states that the diagonal matrix elements vanish, leaving: @@ -49,16 +49,16 @@ $$\begin{aligned} \end{aligned}$$ We now choose $$\hat{H}_1$$ to be as follows, -sinusoidally oscillating with a spatially odd $$V(\vec{r})$$: +sinusoidally oscillating with a spatially odd $$\hat{V}(\vec{r})$$: $$\begin{aligned} \hat{H}_1(t) - = V \cos(\omega t) - = \frac{V}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big) + = \hat{V} \cos(\omega t) + = \frac{\hat{V}}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big) \end{aligned}$$ We insert this into the equations for $$c_a$$ and $$c_b$$, -and define $$V_{ab} \equiv \matrixel{a}{V}{b}$$, leading us to: +and define $$V_{ab} \equiv \matrixel{a}{\hat{V}}{b}$$, leading us to: $$\begin{aligned} \dv{c_a}{t} @@ -124,7 +124,7 @@ which are found to be: $$\begin{aligned} \lambda_1 = i \frac{\omega - \omega_0 + \tilde{\Omega}}{2} - \qquad \quad + \qquad \qquad \lambda_2 = i \frac{\omega - \omega_0 - \tilde{\Omega}}{2} \end{aligned}$$ @@ -140,7 +140,7 @@ $$\begin{aligned} So that the general solution $$c_a(t)$$ is as follows, where $$A$$ and $$B$$ are arbitrary constants, -to be determined from initial conditions (and normalization): +determined by the initial conditions and normalization: $$\begin{aligned} \boxed{ @@ -154,7 +154,7 @@ from the coupled equation we started at, or, if we only care about the probability density $$|c_a|^2$$, we can use $$|c_b|^2 = 1 - |c_a|^2$$. For example, if $$A = 0$$ and $$B = 1$$, -we get the following probabilities +we get the following probabilities: $$\begin{aligned} |c_a(t)|^2 @@ -168,9 +168,8 @@ $$\begin{aligned} Note that the period was halved by squaring. This periodic "flopping" of the particle between $$\ket{a}$$ and $$\ket{b}$$ -is known as **Rabi oscillation**, **Rabi flopping** or the **Rabi cycle**. -This is a more accurate treatment -of the flopping found from first-order +is called **Rabi oscillation**, **Rabi flopping** or the **Rabi cycle**. +This is a more accurate treatment of the flopping found from first-order [time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/). The name **generalized Rabi frequency** suggests @@ -183,11 +182,13 @@ $$\begin{aligned} \equiv \frac{V_{ba}}{\hbar} \end{aligned}$$ -Some authors use $$|V_{ba}|$$ instead, -but not doing that lets us use $$\Omega$$ as a nice abbreviation. -As an example, Rabi oscillation arises -in the [electric dipole approximation](/know/concept/electric-dipole-approximation/), -where $$\hat{H}_1$$ is: +Some authors write $$|V_{ba}|$$ instead, +but not doing so lets us use $$\Omega$$ as a nice abbreviation. +For example, Rabi oscillation arises for electrons +in an oscillating electric field (e.g. a light wave), +in which case $$\hat{H}_1$$ is as follows according to +the [electric dipole approximation](/know/concept/electric-dipole-approximation/), +with charge $$q < 0$$: $$\begin{aligned} \hat{H}_1(t) diff --git a/source/know/concept/time-dependent-perturbation-theory/index.md b/source/know/concept/time-dependent-perturbation-theory/index.md index b4b35e1..841c219 100644 --- a/source/know/concept/time-dependent-perturbation-theory/index.md +++ b/source/know/concept/time-dependent-perturbation-theory/index.md @@ -70,11 +70,11 @@ $$\begin{aligned} \end{aligned}$$ And so forth. The pattern here is clear: we can calculate the $$(j\!+\!1)$$th -correction using only our previous result for the $$j$$th correction. +correction using our result for the $$j$$th correction. The only purpose of $$\lambda$$ was to help us collect its orders; in the end we simply set $$\lambda = 1$$ or absorb it into $$\hat{H}_1$$. Now we have the essence of time-dependent perturbation theory, -we cannot go any further without considering a specific $$\hat{H}_1$$. +and we cannot go any further without considering a specific $$\hat{H}_1$$. |