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author | Prefetch | 2022-10-20 18:25:31 +0200 |
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committer | Prefetch | 2022-10-20 18:25:31 +0200 |
commit | 16555851b6514a736c5c9d8e73de7da7fc9b6288 (patch) | |
tree | 76b8bfd30f8941d0d85365990bcdbc5d0643cabc /source/know/concept/pauli-exclusion-principle/index.md | |
parent | e5b9bce79b68a68ddd2e51daa16d2fea73b84fdb (diff) |
Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'
Diffstat (limited to 'source/know/concept/pauli-exclusion-principle/index.md')
-rw-r--r-- | source/know/concept/pauli-exclusion-principle/index.md | 56 |
1 files changed, 28 insertions, 28 deletions
diff --git a/source/know/concept/pauli-exclusion-principle/index.md b/source/know/concept/pauli-exclusion-principle/index.md index 58a3f69..9821718 100644 --- a/source/know/concept/pauli-exclusion-principle/index.md +++ b/source/know/concept/pauli-exclusion-principle/index.md @@ -12,9 +12,9 @@ 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 +$$\ket{x_1}\ket{x_2} = \ket{x_1} \otimes \ket{x_2}$$, 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: $$\begin{aligned} @@ -28,22 +28,22 @@ $$\begin{aligned} \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} \end{aligned}$$ As it turns out, in nature, each class of particle has a single -associated permutation eigenvalue $\lambda$, or in other words: whether -$\lambda$ is $-1$ or $1$ depends on the type of particle that $x_1$ -and $x_2$ are. Particles with $\lambda = -1$ are called -**fermions**, and those with $\lambda = 1$ are known as **bosons**. We -define $\hat{P}_f$ with $\lambda = -1$ and $\hat{P}_b$ with -$\lambda = 1$, such that: +associated permutation eigenvalue $$\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} @@ -53,20 +53,20 @@ $$\begin{aligned} Another fundamental fact of nature is that identical particles cannot be distinguished by any observation. Therefore it is impossible to tell -apart $\Ket{a}\Ket{b}$ and the permuted state $\Ket{b}\Ket{a}$, -regardless of the eigenvalue $\lambda$. There is no physical difference! +apart $$\Ket{a}\Ket{b}$$ and the permuted state $$\Ket{b}\Ket{a}$$, +regardless of the eigenvalue $$\lambda$$. There is no physical difference! -But this does not mean that $\hat{P}$ is useless: despite not having any +But this does not mean that $$\hat{P}$$ is useless: despite not having any observable effect, the resulting difference between fermions and bosons is absolutely fundamental. Consider the following superposition state, -where $\alpha$ and $\beta$ are unknown: +where $$\alpha$$ and $$\beta$$ are unknown: $$\begin{aligned} \Ket{\Psi(a, b)} = \alpha \Ket{a}\Ket{b} + \beta \Ket{b}\Ket{a} \end{aligned}$$ -When we apply $\hat{P}$, we can "choose" between two "intepretations" of +When we apply $$\hat{P}$$, we can "choose" between two "intepretations" of its action, both shown below. Obviously, since the left-hand sides are equal, the right-hand sides must be equal too: @@ -78,28 +78,28 @@ $$\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) \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) \end{aligned}$$ -This state is called **antisymmetric** under exchange: switching $a$ and $b$ +This state is called **antisymmetric** under exchange: switching $$a$$ and $$b$$ causes a sign change, as we would expect for fermions. -Now, what if the particles $x_1$ and $x_2$ are in the same state $a$? -For bosons, we just need to update the normalization constant $C$: +Now, what if the particles $$x_1$$ and $$x_2$$ are in the same state $$a$$? +For bosons, we just need to update the normalization constant $$C$$: $$\begin{aligned} \Ket{\Psi(a, a)}_b |