25.1 Multiple degrees of freedom

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

Up until now we have been working with the simple case of one physical degree of freedom, $i$ .e., one pair $(Q, P)$ of position and momentum operators satisfying the Heisenberg relation $[Q, P] = i 1$ , or one pair of adjoint operators $a, a^{†}$ satisfying $[a, a^{†}] = 1$ . We can easily extend this to any number $d$ of degrees of freedom by taking tensor products of our state space $F_{:}$ and d copies of our operators, each acting on a diferent factor of the tensor product. Our new state space will be

H = F_{d} = dtimes F \otimes \dots \otimes F

and we will have operators

Q_{j}, P_{j} j = 1, \dots, d

satisfying

[Q_{j}, P_{k}] = i δ_{j k} 1, [Q_{j}, Q_{k}] = [P_{j}, P_{k}] = 0

Here $Q_{j}$ and $P_{j}$ act on the $j^{;}$ ’th term of the tensor product in the usual way, and trivially on the other terms.

We define annihilation and creation operators then by

a_{j} = \frac{1}{2} (Q_{j} + i P_{j}), a_{j}^{†} = \frac{1}{2} (Q_{j} - i P_{j}), j = 1, \dots, d

These satis $fy$ :

Definition (Canonical commutation relations)

The canonical commutation $r e -$ lations $(o f t e n$ abbreviated CCR) are
$[a_{j}, a_{k}^{†}] = δ_{j k} 1, [a_{j}, a_{k}] = [a_{j}^{†}, a_{k}^{†}] = 0$

In the Bargmann-Fock representation $H = F_{d}$ is the space of holomorphic functions in $d$ complex variables $z_{j}$ (with finite norm in the d dimensional version of 22.4) and we have

a_{j} = \frac{\partial}{\partial z _{j}}, a_{j}^{†} = z_{j}

The harmonic oscillator Hamiltonian for $d$ degrees of freedom will be

H = \frac{1}{2} j = 1 \sum d (P_{j}^{2} + Q_{j}^{2}) = j = 1 \sum d (a_{j}^{†} a_{j} + \frac{1}{2}) (25.1)

where one should keep in mind that each degree of freedom can be rescaled separately, allowing diferent parameters $ω$ for the diferent degrees of freedom. The energy and number operator eigenstates will be written

∣ n_{1}, \dots, n_{d} ⟩

where

a_{j}^{†} a_{j} ∣ n_{1}, \dots, n_{d} ⟩ = N_{j} ∣ n_{1}, \dots, n_{d} ⟩ = n_{j} ∣ n_{1}, \dots, n_{d} ⟩

For $d = 3$ the harmonic oscillator problem is an example of the central potential problem described in chapter 21, and will be discussed in more detail in section 25.4.2. It has an $S O (3)$ symmetry, with angular momentum operators that commute with the Hamiltonian, and spaces of energy eigenstates that can be organized into irreducible $S O (3)$ representations. In the Schr¨odinger representation states are in $H = L^{2} (R^{3})$ , described by wavefunctions that can be written in rectangular or spherical coordinates, and the Hamiltonian is a second-order diferential operator. In the Bargmann-Fock representation, states in $F_{3}$ are described by holomorphic functions of 3 complex variables, with operators given in terms of products of annihilation and creation operators. The Hamiltonian is, up to a constant, just the number operator, with energy eigenstates homogeneous polynomials (with eigenvalue of the number operator their degree).

Either the $P_{j}, Q_{j}$ or the $a_{j}, a_{j}^{†}$ together with the identity operator will give a representation of the Heisenberg Lie algebra $h_{2 d + 1}$ on $H,$ and by exponentiation a representation of the Heisenberg group $H_{2 d + 1}$ . Quadratic combinations of these operators will give a representation of the Lie algebra $sp (2 d, R)$ , one that exponentiates to the metaplectic representation of a double cover of $S p (2 d, R)$ .

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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 276、277、278、279、280、281、282、283、284、285、286、287、288、289

来源版本：2025-10-20

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25.1 Multiple degrees of freedom

25.1 Multiple degrees of freedom

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