10.2 Translations in time and space

10.2 Translations in time and space

10.2.1 Energy and the group R of time translations

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

We have seen that it is a basic axiom of quantum mechanics that the observable operator responsible for infinitesimal time translations is the Hamiltonian operator $H$, a fact that is expressed as the Schr¨odinger equation

$$i\hslash \frac{d}{dt}|\psi ⟩=H|\psi ⟩$$

When $H$ is time-independent, this equation can be understood as reflecting the existence of a unitary representation $(U(t),\mathcal{H})$ of the group $R$ of time translations on the state space $\mathcal{H}$

When $H$ is finite dimensional, the fact that a diferentiable unitary representation $U(t)$ of $R$ on $H$ is of the form

$$U(t)=e^{-\frac{i}{\hslash }tH}$$

for H a self-adjoint matrix follows from the same sort of argument as in theorem 2.3. Such a $U(t)$ provides solutions of the Schr¨odinger equation by

$$|\psi (t)⟩=U(t)|\psi (0)⟩$$

The Lie algebra of $R$ is also R and we get a Lie algebra representation of R by taking the time derivative of $U(t)$ , which gives us

$$\hslash \frac{d}{dt}U(t{)}_{|t=0}=-iH$$

Because this Lie algebra representation comes from taking the derivative of a unitary representation, $-iH$ will be skew-adjoint, so H will be self-adjoint.

10.2.2 Momentum and the group R³ of space translations

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

Since we now want to describe quantum systems that depend not just on time, but on space variables $\mathbf{q}=(q_1,q_2,q_3)$ , we will have an action by unitary transformations of not just the group $R$ of time translations, but also the group ${\mathbf{R}}^3$ of spatial translations. We will define the corresponding Lie algebra representations using self-adjoint operators $P_1,P_2,P_3$ that play the same role for spatial translations that the Hamiltonian plays for time translations:

Definition (Momentum operators)

For a quantum system with state space $\mathcal{H}=L^2({\mathbf{R}}^3)$ given by complex-valued functions of position variables $q_1,q_2,q_3$ momentum operators $P_1,P_2,P_3$ are defined by

$$P_1=-i\hslash \frac{\partial }{\partial q_1},P_2=-i\hslash \frac{\partial }{\partial q_2},P_3=-i\hslash \frac{\partial }{\partial q_3}$$

These are given the name “momentum operators” since we will see that their eigenvalues have an interpretation as the components of the momentum vector for the system, just as the eigenvalues of the Hamiltonian have an interpretation as the energy. Note that while in the case of the Hamiltonian the factor of ℏ kept track of the relative normalization of energy and time units, here it plays the same role for momentum and length units. It can be set to one if appropriate choices of units of momentum and length are made.

The diferentiation operator is skew-adjoint since, using integration by parts³ one has for each variable, for ${\psi }_1,{\psi }_2\in \mathcal{H}$

$$\begin{array}{rl}{\int }_{-\infty }^{+\infty }{\overline{\psi }}_1\left(\frac{d}{dq}{\psi }_2\right)dq& ={\int }_{-\infty }^{+\infty }\left(\frac{d}{dq}({\overline{\psi }}_1{\psi }_2)-\left(\frac{d}{dq}{\overline{\psi }}_1\right){\psi }_2\right)dq\\ & =-{\int }_{-\infty }^{+\infty }\left(\frac{d}{dq}{\overline{\psi }}_1\right){\psi }_{2}dq\end{array}$$

(assuming that the ${\psi }_j(q)$ go to $0\mathrm{at}\pm \infty )$ . The $P_j$ are thus self-adjoint operators, with real eigenvalues as expected for an observable operator. Multiplying by −$i$ to get the corresponding skew-adjoint operator of a unitary Lie algebra representation we find

$$-i{P}_j=-\hslash \frac{\partial }{\partial q_j}$$

Up to the ℏ factor that depends on units, these are exactly the Lie algebra representation operators on basis elements of the Lie algebra, for the action of ${\mathbf{R}}^{\bar{3}}$ on functions on ${\mathbf{R}}^3$ induced from translation:

$$\pi (a_1,a_2,a_3)f(q_1,q_2,q_3)=f(q_1-a_1,q_2-a_2,q_3-a_3)$$ $${\pi }^{\prime }(a_1,a_2,a_3)=a_1(-i{P}_1)+a_2(-i{P}_2)+a_3(-i{P}_3)=-\hslash \left(a_1\frac{\partial }{\partial q_1}+a_2\frac{\partial }{\partial q_2}+a_3\frac{\partial }{\partial q_3}\right)$$

Note that the convention for the sign choice here is the opposite from the case of the Hamiltonian $\begin{array}{r}(-iP=-\hslash \frac{d}{da}vs.-iH=\hslash \frac{d}{dt})\end{array}$ . This means that the conventional sign choice we have been using for the Hamiltonian makes it minus the generator of translations in the time direction. The reason for this comes from considerations of special relativity (which will be discussed in chapter 40), where the inner product on space-time has opposite signs for the space and time dimensions.

Previous section: 10.1 The group R and its representations · Next section: 10.3 The energy-momentum relation and the Schr¨odinger equation for a free particle · 中文版本

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