Dynamics I

Before continuing to specific systems, we want to make a couple general comments on dynamics.

Energy-time uncertainty

The position-momentum uncertainty principle is often complemented with an energy-time uncertainty principle:

(269)\[\Delta E \delta t \geq \frac{\hbar}{2}\]

This could be argued for, for example, by noting that the energy is a component of a four-vector for which the spatial momenta are also components, \(p^{\mu} = (E/c, {\vec p})\), and similarly for time and space: \(x^{\mu} = (ct, {\vec x})\). But that isn’t the strongest argument, and at any rate we are dealing with non-relativistic theories. For starters, in this case \(t\) is a parameter not an operator.

Another handwaving argument is that if we measure a system over a finite time \(\Delta t\), the measurement process itself violates time translation invariance. In quantum mechanics as well as in classical mechanics, the conservation of energy follows from this invariance; thus in a measurement \(E\) is no longer conserved.

But we can make a more precise statement that gives some definition of \(\Delta t\). Let \(H\) be the Hamiltonian and \(A\) some Hermitian operator. We have already shown that

(270)\[\begin{split}\begin{align} (\Delta H)^2 (\Delta A)^2 & \geq \frac{1}{4} \Big|\bra{\psi} [H,A]\ket{\psi}\Big|^2\\ & = \frac{1}{4} \Big| i\hbar \bra{\psi}\frac{\del A}{\del t}\ket{\psi}\Big|^2\\ & \Rightarrow \Delta E \left(\frac{\Delta A}{\vev{\frac{\del A}{\del t}}}\right) \geq \frac{\hbar}{2} \end{align}\end{split}\]

If we define \(\Delta H \equiv \Delta E\) and

\[\Delta t = \frac{\Delta A}{\vev{\frac{\del A}{\del t}}}\]

then we get our uncertainty principle. We can ghink of \(\Delta t\) as the time it takes for \(\vev{A}\) to change by an amount \(\Delta A\) – that is, the time scale for a change in the state to be noticeable.

Conservation of probability

We have noted before that \(\psi^*({\vec x}\psi({\vec x})\) is a probability density; one integrates it over a region to find the probability that a particle lives in that region.

Now the total probability in a system must be \(1\). But over time, the probability that a particle is found in a given region can change; however it must change in such a way that the integrated probability does not.

To see how this works, we can use the Schroedinger equation and compute:

(271)\[\begin{split}\begin{align} \frac{\del}{\del t} \psi^*\psi & = \frac{\hbar}{2im} (\nabla^2\psi^*\psi - \psi^* \nabla^2 \psi)\\ & = \vec{\nabla}\cdot\left(\frac{\hbar}{2im}\right)\left({\vec\nabla}\psi^* \psi - \psi^* {\vec{\nabla}}\psi\right) \end{align}\end{split}\]

Note how the potential has dropped out of this equation. If we define the probability current as:

(272)\[{\vec J}_{prob} = \frac{\hbar}{2im} \left(\psi^* {\vec\nabla}\psi - ({\vec\nabla}\psi^*) \psi\right)\]

and the probability density as:

(273)\[\rho = \psi^*\psi\]

then we get the continuity equation

(274)\[\frac{\del \rho}{\del t} + {\vec\nabla}\cdot{\vec J} = 0\]

That is, the change of probability in a region is due to the flux of \({\vec J}\) into and out of that region. This guarantees the conservation of probability in the entire space so long as \({\vec J}\) vanishes quickly enough at infinity.