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6 files changed, 59 insertions, 29 deletions
diff --git a/latex/know/concept/blochs-theorem/source.md b/latex/know/concept/blochs-theorem/source.md index 528c218..79ee9a6 100644 --- a/latex/know/concept/blochs-theorem/source.md +++ b/latex/know/concept/blochs-theorem/source.md @@ -2,7 +2,7 @@ # Bloch's theorem -In quantum mechanics, *Bloch's theorem* states that, +In quantum mechanics, **Bloch's theorem** states that, given a potential $V(\vec{r})$ which is periodic on a lattice, i.e. $V(\vec{r}) = V(\vec{r} + \vec{a})$ for a primitive lattice vector $\vec{a}$, @@ -22,13 +22,38 @@ $$ In other words, in a periodic potential, the solutions are simply plane waves with a periodic modulation, -known as *Bloch functions* or *Bloch states*. +known as **Bloch functions** or **Bloch states**. This is suprisingly easy to prove: if the Hamiltonian $\hat{H}$ is lattice-periodic, -then it will commute with the unitary translation operator $\hat{T}(\vec{a})$, +then both $\psi(\vec{r})$ and $\psi(\vec{r} + \vec{a})$ +are eigenstates with the same energy: + +$$ +\begin{aligned} + \hat{H} \psi(\vec{r}) = E \psi(\vec{r}) + \qquad + \hat{H} \psi(\vec{r} + \vec{a}) = E \psi(\vec{r} + \vec{a}) +\end{aligned} +$$ + +Now define the unitary translation operator $\hat{T}(\vec{a})$ such that +$\psi(\vec{r} + \vec{a}) = \hat{T}(\vec{a}) \psi(\vec{r})$. +From the previous equation, we then know that: + +$$ +\begin{aligned} + \hat{H} \hat{T}(\vec{a}) \psi(\vec{r}) + = E \hat{T}(\vec{a}) \psi(\vec{r}) + = \hat{T}(\vec{a}) \big(E \psi(\vec{r})\big) + = \hat{T}(\vec{a}) \hat{H} \psi(\vec{r}) +\end{aligned} +$$ + +In other words, if $\hat{H}$ is lattice-periodic, +then it will commute with $\hat{T}(\vec{a})$, i.e. $[\hat{H}, \hat{T}(\vec{a})] = 0$. -Therefore $\hat{H}$ and $\hat{T}(\vec{a})$ must share eigenstates $\psi(\vec{r})$: +Consequently, $\hat{H}$ and $\hat{T}(\vec{a})$ must share eigenstates $\psi(\vec{r})$: $$ \begin{aligned} diff --git a/latex/know/concept/dirac-notation/source.md b/latex/know/concept/dirac-notation/source.md index 47aa370..f34047d 100644 --- a/latex/know/concept/dirac-notation/source.md +++ b/latex/know/concept/dirac-notation/source.md @@ -3,18 +3,18 @@ # Dirac notation -*Dirac notation* is a notation to do calculations in a Hilbert space +**Dirac notation** is a notation to do calculations in a Hilbert space without needing to worry about the space's representation. It is basically the *lingua franca* of quantum mechanics. -In Dirac notation there are *kets* $\ket{V}$ from the Hilbert space -$\mathbb{H}$ and *bras* $\bra{V}$ from a dual $\mathbb{H}'$ of the +In Dirac notation there are **kets** $\ket{V}$ from the Hilbert space +$\mathbb{H}$ and **bras** $\bra{V}$ from a dual $\mathbb{H}'$ of the former. Crucially, the bras and kets are from different Hilbert spaces and therefore cannot be added, but every bra has a corresponding ket and vice versa. -Bras and kets can only be combined in two ways: the *inner product* -$\braket{V}{W}$, which returns a scalar, and the *outer product* +Bras and kets can be combined in two ways: the **inner product** +$\braket{V}{W}$, which returns a scalar, and the **outer product** $\ket{V} \bra{W}$, which returns a mapping $\hat{L}$ from kets $\ket{V}$ to other kets $\ket{V'}$, i.e. a linear operator. Recall that the Hilbert inner product must satisfy: diff --git a/latex/know/concept/pauli-exclusion-principle/source.md b/latex/know/concept/pauli-exclusion-principle/source.md index 8870c0c..d1c2149 100644 --- a/latex/know/concept/pauli-exclusion-principle/source.md +++ b/latex/know/concept/pauli-exclusion-principle/source.md @@ -3,7 +3,7 @@ # Pauli exclusion principle -In quantum mechanics, the *Pauli exclusion principle* is a theorem that +In quantum mechanics, the **Pauli exclusion principle** is a theorem that has profound consequences for how the world works. Suppose we have a composite state @@ -26,14 +26,17 @@ $$\begin{aligned} Therefore, $\ket{a}\ket{b}$ is an eigenvector of $\hat{P}^2$ with eigenvalue $1$. Since $[\hat{P}, \hat{P}^2] = 0$, $\ket{a}\ket{b}$ must also be an eigenket of $\hat{P}$ with eigenvalue $\lambda$, -satisfying $\lambda^2 = 1$, so we know that $\lambda = 1$ or -$\lambda = -1$. +satisfying $\lambda^2 = 1$, so we know that $\lambda = 1$ or $\lambda = -1$: + +$$\begin{aligned} + \hat{P} \ket{a}\ket{b} = \lambda \ket{a}\ket{b} +\end{aligned}$$ As it turns out, in nature, each class of particle has a single associated permutation eigenvalue $\lambda$, or in other words: whether $\lambda$ is $-1$ or $1$ depends on the species of particle that $x_1$ and $x_2$ represent. Particles with $\lambda = -1$ are called -*fermions*, and those with $\lambda = 1$ are known as *bosons*. We +**fermions**, and those with $\lambda = 1$ are known as **bosons**. We define $\hat{P}_f$ with $\lambda = -1$ and $\hat{P}_b$ with $\lambda = 1$, such that: @@ -80,14 +83,14 @@ $$\begin{aligned} \end{aligned}$$ Where $C$ is a normalization constant. As expected, this state is -*symmetric*: switching $a$ and $b$ gives the same result. Meanwhile, for +**symmetric**: switching $a$ and $b$ gives the same result. Meanwhile, for fermions ($\lambda = -1$), we find that $\alpha = -\beta$: $$\begin{aligned} \ket{\Psi(a, b)}_f = C \big( \ket{a}\ket{b} - \ket{b}\ket{a} \big) \end{aligned}$$ -This state is called *antisymmetric* under exchange: switching $a$ and $b$ +This state is called **antisymmetric** under exchange: switching $a$ and $b$ causes a sign change, as we would expect for fermions. Now, what if the particles $x_1$ and $x_2$ are in the same state $a$? @@ -106,7 +109,7 @@ $$\begin{aligned} = 0 \end{aligned}$$ -At last, 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 an atom only fit a limited number of -electrons, since each must have a different quantum number. +At last, 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/latex/know/concept/probability-current/source.md b/latex/know/concept/probability-current/source.md index bffc599..bb84121 100644 --- a/latex/know/concept/probability-current/source.md +++ b/latex/know/concept/probability-current/source.md @@ -3,7 +3,7 @@ # Probability current -In quantum mechanics, the *probability current* describes the movement +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: diff --git a/latex/know/concept/time-independent-perturbation-theory/source.md b/latex/know/concept/time-independent-perturbation-theory/source.md index a3167cd..48504f4 100644 --- a/latex/know/concept/time-independent-perturbation-theory/source.md +++ b/latex/know/concept/time-independent-perturbation-theory/source.md @@ -3,8 +3,8 @@ # Time-independent perturbation theory -*Time-independent perturbation theory*, sometimes also called -*stationary state perturbation theory*, is a specific application of +**Time-independent perturbation theory**, sometimes also called +**stationary state perturbation theory**, is a specific application of perturbation theory to the time-independent Schrödinger equation in quantum physics, for Hamiltonians of the following form: @@ -27,8 +27,8 @@ $$\begin{aligned} &= E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ... \end{aligned}$$ -Where $E_n^{(1)}$ and $\ket*{\psi_n^{(1)}}$ are called the *first-order -corrections*, and so on for higher orders. We insert this into the +Where $E_n^{(1)}$ and $\ket*{\psi_n^{(1)}}$ are called the **first-order +corrections**, and so on for higher orders. We insert this into the Schrödinger equation: $$\begin{aligned} @@ -72,6 +72,7 @@ $$\begin{aligned} The approach to solving the other two equations varies depending on whether this $\hat{H}_0$ has a degenerate spectrum or not. + ## Without degeneracy We start by assuming that there is no degeneracy, in other words, each @@ -178,6 +179,7 @@ $$\begin{aligned} In practice, it is not particulary useful to calculate more corrections. + ## With degeneracy If $\varepsilon_n$ is $D$-fold degenerate, then its eigenstate could be @@ -295,7 +297,7 @@ The trick is to find a Hermitian operator $\hat{L}$ (usually using symmetries of the system) which commutes with both $\hat{H}_0$ and $\hat{H}_1$: $$\begin{aligned} - [\hat{L}, \hat{H}_0] = [\hat{L}, \hat{H}_1] = 0 + \comm*{\hat{L}}{\hat{H}_0} = \comm{\hat{L}}{\hat{H}_1} = 0 \end{aligned}$$ So that it shares its eigenstates with $\hat{H}_0$ (and $\hat{H}_1$), diff --git a/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md b/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md index a50302c..79f344a 100644 --- a/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md +++ b/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md @@ -3,10 +3,10 @@ # Wentzel-Kramers-Brillouin approximation -In quantum mechanics, the *Wentzel-Kramers-Brillouin* or simply the *WKB -approximation* is a method to approximate the wave function $\psi(x)$ of +In quantum mechanics, the **Wentzel-Kramers-Brillouin** or simply the **WKB +approximation** is a method 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 +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 $V(x)$ @@ -164,7 +164,7 @@ $$\begin{aligned} What if $E < V$? In classical mechanics, this is not allowed; a ball cannot simply go through a potential bump without the necessary energy. -However, in quantum mechanics, particles can *tunnel* through barriers. +However, in quantum mechanics, particles can **tunnel** through barriers. Conveniently, all we need to change for the WKB approximation is to let the momentum take imaginary values: |