36.2 Multi-particle quantum systems of free particles: finite cutof formalism

36.2 Multi-particle quantum systems of free particles: finite cutof formalism

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue

To describe multi-particle quantum systems in terms of quanta of a harmonic oscillator system, we would like to proceed as described in section 36.1, taking solutions to the free particle Schr¨odinger equation (discussed in chapters 10 and 11) as the single-particle state space. Recall that for a free particle in one spatial dimension such solutions are given by complex-valued functions on $R$, with observables the self-adjoint operators for momentum

$$P=-i\frac{d}{dx}$$

and energy (the Hamiltonian)

$$H=\frac{P^2}{2m}=-\frac{1}{2m}\frac{d^2}{d{x}^2}$$

Eigenfunctions for both P and H are the functions of the form

$${\psi }_p(x)\propto e^{ipx}$$

for $p\in \mathbf{R}.$ , with eigenvalues $p$ for $P$ and $\frac{p^2}{2m}$ for $H$ . Recall that these eigenfunctions are not normalizable, and thus not in the conventional choice of state space as $L^2(\mathbf{R})$ ).

As we saw in section 11.1, one way to deal with this issue is to do what physicists sometimes refer to as “putting the system in a box”, by imposing periodic boundary conditions

$$\psi (x+L)=\psi (x)$$

for some number $L,$ efectively restricting the relevant values of x to be considered to those on an interval of length $L.$ For our eigenfunctions, this condition is

$$e^{ip(x+L)}=e^{ipx}$$

so we must have

$$e^{ipL}=1$$

which implies that

$$p=\frac{2\pi }{L}j\equiv p_j$$

for $j$ an integer. Then the momentum will take on a countable number of discrete values corresponding to the $j\in \mathbf{Z},$ , and

$$|j⟩={\psi }_j(x)=\frac{1}{\sqrt{L}}{e}^{i{p}_{j}x}=\frac{1}{\sqrt{L}}{e}^{i\frac{2\pi j}{L}x}$$

will be orthonormal eigenfunctions satisfying

$$⟨j^{\prime }|j⟩={\delta }_{j{j}^{\prime }}$$

This use of periodic boundary conditions is one form of what physicists call an “infrared ${\mathrm{cutoff}}^{,}$ , a way of removing degrees of freedom that correspond to arbitrarily large sizes, in order to make the quantum system well-defined. One starts with a fixed value of $L,$ and only later studies the limit $L\to \infty$

The number of degrees of freedom is now countable, but still infinite, and something more must be done in order to make the single-particle state space finite dimensional. This can be accomplished with an additional cutof, an “ultraviolet cutof”, which means restricting attention to $|p|\le \Lambda$ for some finite $\Lambda ,$ or equivalently $\begin{array}{r}|j|<\frac{\Lambda L}{2\pi }\end{array}$ . This makes the space of solutions finite dimensional, allowing quantization by use of the Bargmann-Fock method used for the finite dimensional harmonic oscillator. The $\Lambda \infty$ and $L\infty$ limits can then be taken at the end of a calculation.

The Schr¨odinger equation is a first-order diferential equation in time, $t$ and solutions can be completely characterized by their initial value at $t=0$

$$\psi (x,0)=\sum _{j=-\frac{\Lambda L}{2\pi }}^{+\frac{\Lambda L}{2\pi }}\alpha (p_j)e^{i\frac{2\pi j}{L}x}$$

determined by a choice of complex coeficients $\alpha =\{\alpha (p_j)\}$ . At later times the solution will be given by

$$\psi (x,t)=\sum _{j=-\frac{\Lambda L}{2\pi }}^{+\frac{\Lambda L}{2\pi }}\alpha (p_j)e^{i{p}_{j}x}{e}^{-i\frac{p_j^2}{2m}t}\text{\hspace{1em}(36.5)}$$

Our space of solutions is the space of all sets of complex numbers $\alpha (p_j)$ . In principle we could take this space as our dual phase space and quantize using the Schr¨odinger representation, for instance taking the real parts of the $\alpha (p_j)$ as position-like coordinates. Especially since our dual phase space is already complex, it is much more convenient to use the Bargmann-Fock method of quantization. Recalling the discussion of section 26.4, we will need both a dual phase space $\mathcal{M}$ and its conjugate space $\overline{\mathcal{M}}$ , which means that we will need to consider not just solutions of the Schr¨odinger equation, but of its conjugate

$$-i\frac{\partial }{\partial t}\overline{\psi }=-\frac{1}{2m}\frac{{\partial }^2}{\partial x^2}\overline{\psi }\text{\hspace{1em}(36.6)}$$

which is satisfied by conjugates $\overline{\psi }$ of solutions $\psi$ of the usual Schr¨odinger equation. We will take $\mathcal{M}={\mathcal{H}}_1$ to be the space of Schr¨odinger equation solutions 36.5. $\overline{\mathcal{M}}={\overline{\mathcal{H}}}_1$ will be the space of solutions of 36.6, which can be written

$$\sum _{j=-\frac{\Lambda L}{2\pi }}^{+\frac{\Lambda L}{2\pi }}\overline{\alpha }(p_j)e^{-i{p}_{j}x}{e}^{i\frac{p_j^2}{2m}t}$$

for some complex numbers $\overline{\alpha }({\mathbf{p}}_j)$ .

A basis for M will be given by the

$$A(p_j)=\{\begin{array}{ll}\alpha (p_k)=0& k\ne j\\ \alpha (p_k)=1& k=j\end{array}$$

with conjugates $\overline{A}(p_j)$ a basis for $\overline{\mathcal{M}}$ . The Poisson bracket on $\mathcal{M}\oplus \overline{\mathcal{M}}$ will be determined by the following Poisson bracket relations on basis elements

$$\{A(p_j),A(p_k)\}=\{\overline{A}(p_j),\overline{A}(p_k)\}=0,\{A(p_j),\overline{A}(p_k)\}=i{\delta }_{jk}\text{\hspace{1em}(36.7)}$$

Bargmann-Fock quantization gives as state space a Fock space ${\mathcal{F}}_D$ , where $D$ is the number of values of $p_j$ . This is (ignoring issues of completion) the space of polynomials in the $D$ variables $A(p_j)$ ). One has a pair of annihilation and creation operators

$$a(p_j)=\frac{\partial }{\partial A(p_j)},a(p_j{)}^{†}=A(p_j)$$

for each possible value of $j,$ which indexes the possible values $p_j$ . These operators satisfy the commutation relations

$$[a(p_j),a(p_k{)}^{†}]={\delta }_{jk}$$

In the occupation number representation of the Fock space, orthonormal basis elements are

$$|\dots ,n_{p_{j-1}},n_{p_j},n_{p_{j+1}},\dots ⟩$$

with annihilation and creation operators acting by

$$a_{p_j}|\dots ,n_{p_{j-1}},n_{p_j},n_{p_{j+1}},\dots ⟩=\sqrt{n_{p_j}}|\dots ,n_{p_{j-1}},n_{p_j}-1,n_{p_{j+1}},\dots ⟩$$ $$a_{p_j}^{†}|\dots ,n_{p_{j-1}},n_{p_j},n_{p_{j+1}},\dots ⟩=\sqrt{n_{p_j}+1}|\dots ,n_{p_{j-1}},n_{p_j}+1,n_{p_{j+1}},\dots ⟩$$

The occupation number $n_{p_j}$ is the eigenvalue of the operator $a(p_j{)}^{†}a(p_j)$ and takes values $0,1,2,\cdots ,\infty$ . It has a physical interpretation as the number of particles in the state with momentum $p_j$ (recall that such momentum values are discretized in units of $\frac{2\pi }{L}$ , and in the interval $[-\Lambda ,\Lambda ])$ . The state with all occupation numbers equal to zero is denoted

$$|\dots ,0,0,0,\dots ⟩=|0⟩$$

and called the “vacuum” state.

Observables that can be built out of the annihilation and creation operators include

• The total number operator

$$\hat{N}=\sum _{k}a(p_k{)}^{†}a(p_k)\text{\hspace{1em}(36.8)}$$

which will have as eigenvalues the total number of particles

$$\hat{N}|\dots ,n_{p_{j-1}},n_{p_j},n_{p_{j+1}},\dots ⟩=(\sum _k{n}_{p_k})|\dots ,n_{p_{j-1}},n_{p_j},n_{p_{j+1}},\dots ⟩$$

• The momentum operator

$$\hat{P}=\sum _k{p}_{k}a(p_k{)}^{†}a(p_k)\text{\hspace{1em}(36.9)}$$

with eigenvalues the total momentum of the multi-particle system.

$$\hat{P}|\dots ,n_{p_{j-1}},n_{p_j},n_{p_{j+1}},\dots ⟩=(\sum _k{n}_{p_k}{p}_k)|\dots ,n_{p_{j-1}},n_{p_j},n_{p_{j+1}},\dots ⟩$$

• The Hamiltonian

$$\hat{H}=\sum _k\frac{p_k^2}{2m}a(p_k{)}^{†}a(p_k)\text{\hspace{1em}(36.10)}$$

which has eigenvalues the total energy

$$\hat{H}|\dots ,n_{p_{j-1}},n_{p_j},n_{p_{j+1}},\dots ⟩=\left(\sum _k{n}_{p_k}\frac{p_k^2}{2m}\right)|\dots ,n_{p_{j-1}},n_{p_j},n_{p_{j+1}},\dots ⟩$$

With ultraviolet and infrared cutofs in place, the possible values of $p_j$ are of a finite number $D$ which is also the complex dimension of ${\mathcal{H}}_1$ . The Hamiltonian operator is the standard harmonic oscillator Hamiltonian, with diferent frequencies

$${\omega }_j=\frac{p_j^2}{2m}$$

for diferent values of $j.$ . Note that we are using normal ordered operators here, which is necessary since in the limit as one or both cutofs are removed, ${\mathcal{H}}_1$ becomes infinite dimensional, and only the normal ordered version of the Hamiltonian used here is well-defined (the non-normal ordered version will difer by an infinite sum of $\frac{1}{2}\mathrm{s})$

Everything in this section has a straightforward analog describing a multiparticle system of fermionic particles with energy-momentum relation given by the free particle Schr¨odinger equation. The annihilation and creation operators will be the fermionic ones, satisfying the canonical anticommutation relations

$$[a_{F{p}_j},a_{F{p}_k}^{†}{]}_{+}={\delta }_{jk}$$

implying that states will have occupation numbers $n_{p_j}=0,1$ , automatically implementing the Pauli principle.

Previous section: 36.1 Multi-particle quantum systems as quanta of a harmonic oscillator · Next section: 36.3 Continuum formalism · 中文版本

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 379、380、381、382、383、384、385、386、387、388、389、390、391、392、393

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：711120ffa7830d5f979b82a637d70aba919820a16ef311f17a47d45c06750e73

OCR 产物 SHA-256：711120ffa7830d5f979b82a637d70aba919820a16ef311f17a47d45c06750e73