11 Fourier Analysis and the Free Particle

The quantum theory of a free particle requires not just a state space $\mathcal{H},$ but also an inner product on $H$, which should be translation invariant so that translations act as unitary transformations. Such an inner product will be given by the integral

$$⟨{\psi }_1,{\psi }_2⟩=C{\int }_{{\mathbf{R}}^3}\overline{{\psi }_1(\mathbf{q})}{\psi }_2(\mathbf{q})d^3\mathbf{q}$$

for some choice of normalization constant $C,$ , usually taken to be $C=1$ . H will be the space $L^2({\mathbf{R}}^3)$ of square-integrable complex-valued functions on ${\mathbf{R}}^3$

A problem arises though if we try and compute the norm-squared of one of our momentum eigenstates |k⟩. We find

$$⟨\mathbf{k}|\mathbf{k}⟩=C{\int }_{{\mathbf{R}}^3}(e^{-i\mathbf{k}\cdot \mathbf{q}})(e^{i\mathbf{k}\cdot \mathbf{q}})d^3\mathbf{q}=C{\int }_{{\mathbf{R}}^3}1d^3\mathbf{q}=\infty$$

As a result there is no value of C which will give these states a finite norm, and they are not in the expected state space. The finite dimensional spectral theorem 4.1 assuring us that, given a self-adjoint operator, we can find an orthonormal basis of its eigenvectors, will no longer hold. Other problems arise because our momentum operators $P_j$ may take states in $H$ to states that are not in $\mathcal{H}(i.e.$ 2 not square-integrable).

We’ll consider two diferent ways of dealing with these problems, for simplicity treating the case of just one spatial dimension. In the first, we impose periodic boundary conditions, efectively turning space into a circle of finite extent, leaving for later the issue of taking the size of the circle to infinity. The translation group action then becomes the $U(1)$ group action of rotation about the circle. This acts on the state space $\mathcal{H}=L^2(S^1)$ , a situation which can be analyzed using the theory of Fourier series. Momentum eigenstates are now in $H$, and labeled by an integer.

While this deals with the problem of eigenvectors not being in $\mathcal{H},$ it ruins an important geometrical structure of the free particle quantum system, by treating positions (taking values in the circle) and momenta (taking values in the integers) quite diferently. In later chapters we will see that physical systems like the free particle are best studied by treating positions and momenta as realvalued coordinates on a single vector space, called phase space. To do this, a formalism is needed that treats momenta as real-valued variables on a par with position variables.

The theory of Fourier analysis provides the required formalism, with the Fourier transform interchanging a state space $L^2(\mathbf{R})$ of wavefunctions depending on position with a unitarily equivalent one using wavefunctions that depend on momenta. The problems of the domain of the momentum operator $P$ and its eigenfunctions not being in $L^2(\mathbf{R})$ still need to be addressed. This can be done by introducing

• a space $\mathcal{S}(\mathbf{R})\subset {\mathcal{L}}^2(\mathbf{R})$ of suficiently well-behaved functions on which $P$ is well-defined, and

• a space $S^{\prime }(\mathbf{R})\supset L^2(\mathbf{R})$ of “generalized functions”, also known as distributions, which will include the eigenvectors of $P.$

Solutions to the Schr¨odinger equation can be studied in any of the three

$$S(\mathbf{R})\subset L^2(\mathbf{R})\subset {\mathcal{S}}^{\prime }(\mathbf{R})$$

contexts, each of which will be preserved by the Fourier transform and allow one to treat position and momentum variables on the same footing.

中文目录

Chapter contents

• 11.1 Periodic boundary conditions and the group U(1)
• 11.2 The group R and the Fourier transform
• 11.3 Distributions
• 11.4 Linear transformations and distributions
• 11.5 Solutions of the Schr¨odinger equation in momentum space
• 11.6 For further reading

Previous chapter: 10 Momentum and the Free Particle

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正文：英文 · OCR 机器稿 · 待校对

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来源版本：2025-10-20

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