Schrödinger's equations of motion

Up to now, we have only considered systems at one particular instant of time. Let us now investigate how quantum mechanical systems evolve with time.

Consider a system in a state $A$ which evolves in time. At time $t$ the state of the system is represented by the ket $\vert At\rangle$. The label $A$ is needed to distinguish the ket from any other ket ($\vert Bt\rangle$, say) which is evolving in time. The label $t$ is needed to distinguish the different states of the system at different times.

The final state of the system at time $t$ is completely determined by its initial state at time $t_0$ plus the time interval $t-t_0$ (assuming that the system is left undisturbed during this time interval). However, the final state only determines the direction of the final state ket. Even if we adopt the convention that all state kets have unit norms, the final ket is still not completely determined, since it can be multiplied by an arbitrary phase-factor. However, we expect that if a superposition relation holds for certain states at time $t_0$ then the same relation should hold between the corresponding time-evolved states at time $t$, assuming that the system is left undisturbed between times $t_0$ and $t$. In other words, if

$$\vert Rt_0\rangle = \vert At_0\rangle + \vert B t_0\rangle$$ (219)

for any three kets, then we should have

$$\vert Rt\rangle = \vert At\rangle + \vert B t\rangle.$$ (220)

This rule determines the time-evolved kets to within a single arbitrary phase-factor to be multiplied into all of them. The evolved kets cannot be multiplied by individual phase-factors, since this would invalidate the superposition relation at later times.

According to Eqs. (219) and (220), the final ket $\vert Rt\rangle$ depends linearly on the initial ket $\vert Rt_0\rangle$. Thus, the final ket can be regarded as the result of some linear operator acting on the initial ket: , i.e.,

$$\vert Rt\rangle = T \vert Rt_0\rangle,$$ (221)

where $T$ is a linear operator which depends only on the times $t$ and $t_0$. The arbitrary phase-factor by which all time evolved kets may be multiplied results in $T(t, t_0)$ being undetermined to an arbitrary multiplicative constant of modulus unity.

Since we have adopted a convention in which the norm of any state ket is unity, it make sense to define the time evolution operator $T$ in such a manner that it preserves the length of any ket upon which it acts (i.e., if a ket is properly normalized at time $t$ then it will remain normalized at all subsequent times $t>t_0$). This is always possible, since the length of a ket possesses no physical significance. Thus, we require that

$$\langle A t_0\vert A t_0\rangle =\langle A t\vert A t\rangle$$ (222)

for any ket $A$, which immediately yields

$$T^{\dag } T = 1.$$ (223)

Hence, the time evolution operator $T$ is a unitary operator.

Up to now, the time evolution operator $T$ looks very much like the spatial displacement operator $D$ introduced in the previous section. However, there are some important differences between time evolution and spatial displacement. In general, we do expect the expectation value of some observable $\xi$ to evolve with time, even if the system is left in a state of undisturbed motion (after all, time evolution has no meaning unless something observable changes with time). The triple product $\langle A\vert\xi\vert A\rangle$ can evolve either because the ket $\vert A\rangle$ evolves and the operator $\xi$ stays constant, the ket $\vert A\rangle$ stays constant and the operator $\xi$ evolves, or both the ket $\vert A\rangle$ and the operator $\xi$ evolve. Since we are already committed to evolving state kets, according to Eq. (221), let us assume that the time evolution operator $T$ can be chosen in such a manner that the operators representing the dynamical variables of the system do not evolve in time (unless they contain some specific time dependence).

We expect, from physical continuity, that as $t\rightarrow t_0$ then $\vert At\rangle\rightarrow \vert A t_0\rangle$ for any ket $A$. Thus, the limit

$$\lim_{t\rightarrow t_0} \frac{\vert At\rangle - \vert At_0\r... ...0} = \lim_{t\rightarrow t_0}\frac{T-1}{t-t_0}\vert At_0\rangle$$ (224)

should exist. Note that this limit is simply the derivative of $\vert A t_0\rangle$ with respect to $t_0$. Let

$$\tau(t_0) = \lim_{t\rightarrow t_0}\frac{T(t, t_0)-1}{t-t_0}.$$ (225)

It is easily demonstrated from Eq. (223) that $\tau$ is anti-Hermitian: i.e.,

$$\tau^{\dag } + \tau = 0.$$ (226)

The fact that $T$ can be replaced by $T\exp({\rm i} \gamma)$ (where $\gamma$ is real) implies that $\tau$ is undetermined to an arbitrary imaginary additive constant (see previous section). Let us define the Hermitian operator $H(t_0)= {\rm i} \hbar \tau$. This operator is undetermined to an arbitrary real additive constant. It follows from Eqs. (224) and (225) that

$${\rm i} \hbar \frac{d\vert At_0\rangle}{dt_0} = {\rm i} ... ...,\hbar \tau(t_0)\vert At_0\rangle = H(t_0) \vert At_0\rangle.$$ (227)

When written for general $t$ this equation becomes

$${\rm i} \hbar \frac{d\vert At\rangle}{dt} = H(t)\vert At\rangle.$$ (228)

Equation (228) gives the general law for the time evolution of a state ket in a scheme in which the operators representing the dynamical variables remain fixed. This equation is denoted Schrödinger's equation of motion. It involves a Hermitian operator $H(t)$ which is, presumably, a characteristic of the dynamical system under investigation.

We saw, in the previous section, that if the operator $D(x, x_0)$ displaces the system along the $x$-axis from $x_0$ to $x$ then

$$p_x = {\rm i} \hbar \lim_{x\rightarrow x_0} \frac{D(x,x_0)-1}{x-x_0},$$ (229)

where $p_x$ is the operator representing the momentum conjugate to $x$. We now have that if the operator $T(t, t_0)$ evolves the system in time from $t_0$ to $t$ then

$$H(t_0) = {\rm i} \hbar \lim_{t\rightarrow t_0} \frac{T(t, t_0)-1}{t-t_0}.$$ (230)

Thus, the dynamical variable corresponding to the operator $H$ stands to time $t$ as the momentum $p_x$ stands to the coordinate $x$. By analogy with classical physics, this suggests that $H(t)$ is the operator representing the total energy of the system. (Recall that, in classical physics, if the equations of motion of a system are invariant under an $x$-displacement of the system then this implies that the system conserves momentum in the $x$-direction. Likewise, if the equations of motion are invariant under a temporal displacement then this implies that the system conserves energy.) The operator $H(t)$ is usually called the Hamiltonian of the system. The fact that the Hamiltonian is undetermined to an arbitrary real additive constant is related to the well-known phenomenon that energy is undetermined to an arbitrary additive constant in physics (i.e., the zero of potential energy is not well-defined).

Substituting $\vert At\rangle = T \vert At_0\rangle$ into Eq. (228) yields

$${\rm i} \hbar \frac{d T}{dt}\vert At_0\rangle = H(t) T\vert At_0\rangle.$$ (231)

Since this must hold for any initial state $\vert A t_0\rangle$ we conclude that

$${\rm i} \hbar \frac{dT}{dt} = H(t) T.$$ (232)

This equation can be integrated to give

$$T(t, t_0) = \exp\left(-{\rm i} \int_{t_0}^t H(t') dt'/\hbar\right),$$ (233)

where use has been made of Eqs. (223) and (224). (Here, we assume that Hamiltonian operators evaluated at different times commute with one another). It is now clear how the fact that $H$ is undetermined to an arbitrary real additive constant leaves $T$ undetermined to a phase-factor. Note that, in the above analysis, time is not an operator (we cannot observe time, as such), it is just a parameter (or, more accurately, a continuous label). Since we are only dealing with non-relativistic quantum mechanics, the fact that position is an operator, but time is only a label, need not worry us unduly. In relativistic quantum mechanics, time and space coordinates are treated on the same footing by relegating position from being an operator to being just a label.