11.1 Periodic boundary conditions and the group U(1)

11.1 Periodic boundary conditions and the group U(1)

Concept links · terms present in this machine draft; source roles are unverified: vector space · complex inner product · eigenvalue · Lie algebra · Lie algebra representation · irreducible representation · U(1)

In this section we’ll describe one way to deal with the problems caused by nonnormalizable eigenstates, considering first the simplified case of a single spatial dimension. In this one dimensional case, the space $R$ is replaced by the circle $S^1$ . This is equivalent to the physicist’s method of imposing “periodic boundary conditions”, meaning to define the theory on an interval, and then identify the ends of the interval. The position variable $q$ can then be thought of as an angle $\varphi$ and one can define the inner product as

$$⟨{\psi }_1,{\psi }_2⟩=\frac{1}{2\pi }{\int }_0^{2\pi }\overline{{\psi }_1(\varphi )}{\psi }_2(\varphi )d\varphi$$

The state space is then

$$\mathcal{H}=L^2(S^1)$$

the space of complex-valued square-integrable functions on the circle.

Instead of the group $R$ acting on itself by translations, we have the standard rotation action of the group $SO(2)$ on the circle. Elements $g(\theta )$ of the group are rotations of the circle counterclockwise by an angle $\theta$, or if we parametrize the circle by an angle $\varphi$, just shifts

$$\varphi \to \varphi +\theta$$

By the same argument as in the case $G=\mathbf{R}.$ , we can use the representation on functions given by equation 1.3 to get a representation on $\mathcal{H}$

$$\pi (g(\theta ))\psi (\varphi )=\psi (\varphi -\theta )$$

If $X$ is a basis of the Lie algebra so(2) (for instance taking the circle as the unit circle in ${\mathbf{R}}^2$ , rotations 2 by 2 matrices, $\begin{array}{c}X=\left(\genfrac{}{}{0ex}{}{0}{1}\right)0\end{array},g(\theta )=e^{\theta X})$ then the Lie algebra representation is given by taking the derivative

$${\pi }^{\prime }(aX)f(\varphi )=\frac{d}{d\theta }f(\varphi -a\theta {)}_{|\theta =0}=-a\frac{d}{d\varphi }f(\varphi )$$

so we have (as in the R case, see equation 10.2)

$${\pi }^{\prime }(aX)=-a\frac{d}{d\varphi }$$

This operator is defined on a dense subspace of $\mathcal{H}=L^2(S^1)$ and is skew-adjoint, since (using integration by parts)

$$\begin{array}{rl}⟨{\psi }_1,\frac{d}{d\varphi }{\psi }_2⟩& =\frac{1}{2\pi }{\int }_0^{2\pi }\overline{{\psi }_1}\frac{d}{d\varphi }{\psi }_{2}d\varphi \\ & =\frac{1}{2\pi }{\int }_0^{2\pi }\left(\frac{d}{d\varphi }(\overline{{\psi }_1}{\psi }_2)-\left(\frac{d}{d\varphi }\overline{{\psi }_1}\right){\psi }_2\right)d\varphi \\ & =-⟨\frac{d}{d\varphi }{\psi }_1,{\psi }_2⟩\end{array}$$

The eigenfunctions of ${\pi }^{\prime }(X)$ are the $e^{in\varphi }$ , for $\mathbf{n}\in \mathbf{Z}$ , which we will also write as state vectors |n⟩. These are orthonormal

$$⟨n|m⟩={\delta }_{nm}\text{\hspace{1em}(11.1)}$$

and provide a countable basis for the space $L^2(S^1)$ . This basis corresponds to the decomposition into irreducibles of H as a representation of $SO(2)$ described above. One has

$$(\pi ,L^2(S^1))={\oplus }_{n\in \mathbf{Z}}({\pi }_n,\mathbf{C})\text{\hspace{1em}(11.2)}$$

where ${\pi }_n$ are the irreducible one dimensional representations given by the multiplication action

$${\pi }_n(g(\theta ))=e^{in\theta }$$

The theory of Fourier series for functions on $S^1$ says that any function $\psi \in$ $L^2(S^1)$ can be expanded in terms of this basis:

Theorem 11.1 (Fourier series)

$If\psi \in L^2(S^1)$ , then

$$|\psi ⟩=\psi (\varphi )=\sum _{n=-\infty }^{+\infty }{c}_n{e}^{in\varphi }=\sum _{n=-\infty }^{+\infty }{c}_n|n⟩$$

where

$$c_n=⟨n|\psi ⟩=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{-in\varphi }\psi (\varphi )d\varphi$$

This is an equality in the sense of the norm on $L^2(S^1)$ , $i$.e.,

$$\underset{N\to \infty }{\lim }||\psi -\sum _{n=-N}^{+N}{c}_n{e}^{in\varphi }||=0$$

The condition that $\psi \in L^2(S^1)$ corresponds to the condition

$$\sum _{n=-\infty }^{+\infty }|c_n{|}^2<\infty$$

on the coeficients $c_n$

One can easily derive the formula for $c_n$ using orthogonality of the $|n⟩$ . For a detailed proof of the theorem see for instance [27] and [84]. The theorem gives an equivalence (as complex vector spaces with a Hermitian inner product) between square-integrable functions on ${\bar{S}}^1$ and square-summable functions on $Z$. As unitary $SO(2)$ representations this is the equivalence of equation 11.2.

The Lie algebra of the group $S^1$ is the same as that of the additive group $R$, and the ${\pi }^{\prime }(X)$ we have found for the $S^1$ action on functions is related to the momentum operator in the same way as in the R case. $\mathrm{So}.$ we can use the same momentum operator

$$P=-i\hslash \frac{d}{d\varphi }$$

which satisfies

$$P|n⟩=\hslash n|n⟩$$

By changing space from the non-compact R to the compact $S^1$ we now have momenta that instead of taking on any real value, can only be integral numbers times ℏ.

Solving the Schr¨odinger equation

$$i\hslash \frac{\partial }{\partial t}\psi (\varphi ,t)=\frac{P^2}{2m}\psi (\varphi ,t)=\frac{-{\hslash }^2}{2m}\frac{{\partial }^2}{\partial {\varphi }^2}\psi (\varphi ,t)$$

as before, we find

$$E{\psi }_E(\varphi )=\frac{-{\hslash }^2}{2m}\frac{d^2}{d{\varphi }^2}{\psi }_E(\varphi )$$

as the eigenvector equation. This has an orthonormal basis of solutions $|n⟩$ with

$$E=\frac{{\hslash }^2{n}^2}{2m}$$

The Schr¨odinger equation is first-order in time, and the space of possible solutions can be identified with the space of possible initial values at a fixed time. Elements of this space of solutions can be characterized by

• The complex-valued square-integrable function $\psi (\varphi ,0)\in L^2(S^1)$ , a function on the circle $S^1$

• The square-summable sequence $c_n$ of complex numbers, a function on the integers $\mathbf{Z}.$

The $c_n$ can be determined from the $\psi (\varphi ,0)$ using the Fourier coeficient formula

$$c_n=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{-in\varphi }\psi (\varphi ,0)d\varphi$$

Given the $c_n$ , the corresponding solution to the Schr¨odinger equation will be

$$\psi (\varphi ,t)=\sum _{n=-\infty }^{+\infty }{c}_n{e}^{in\varphi }{e}^{-i\frac{\hslash n^2}{2m}t}$$

To get something more realistic, we need to take our circle to have an arbitrary circumference $L,$ and we can study our original problem with space $R$ by considering the limit $L\to \infty$ . To do this, we just need to change variables from $\varphi$ to ${\varphi }_L$ , where

$${\varphi }_L=\frac{L}{2\pi }\varphi$$

The momentum operator will now be

$$P=-i\hslash \frac{d}{d{\varphi }_L}$$

and its eigenvalues will be quantized in units of $\frac{2\pi \hslash }{L}$ . The energy eigenvalues will be

$$E=\frac{2{\pi }^2{\hslash }^2{n}^2}{m{L}^2}$$

Note that these values are discrete (as long as the size L of the circle is finite) and non-negative.

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