2.7 Conservation Laws

In classical mechanics, and especially in advanced problem solving techniques, conserved quantities are of upmost importance because of their ability to simplify problems. A conserved quantity is defined classically as one with a time derivative of zero. Unfortunately, in quantum mechanics we cannot really define conserved quantities in the same way, since a particle’s state is fundamentally uncertain until measurement. What we can look at is the time derivative of expectation values.

Let $\hat{O}$ be a Hermitian operator, and assume that it does not explicitly depend on time, i.e. $\frac{\partial\hat{O}}{\partial t}=0$ . Then the expectation value, which may depend on time through the time dependence of the state $\lvert\psi\rangle$ , is

	$\displaystyle\frac{\operatorname{\mathrm{d}}\!}{\operatorname{\mathrm{d}}\!t}% \langle\hat{O}\rangle$	$\displaystyle=\frac{\operatorname{\mathrm{d}}\!}{\operatorname{\mathrm{d}}\!t}% \langle\psi\mathclose{}\|\mathopen{}\hat{O}\mathclose{}\|\mathopen{}\psi\rangle$		(2.147)
		$\displaystyle=\frac{\partial}{\partial t}(\langle\psi\rvert)\hat{O}\lvert\psi% \rangle+\langle\psi\rvert\hat{O}\frac{\partial}{\partial t}\lvert\psi\rangle,$		(2.148)

where in the second line the first partial derivative only acts on the bra. Now we will use the same trick we have used several times already to get rid of the time derivatives by substituting in the Schrodinger equation and its complex conjugate. This time we will substitute in the TDSE. In Dirac notation, the substitution looks like:

\frac{\partial\lvert\psi\rangle}{\partial t}=-\frac{i}{\hbar}\hat{H}\lvert\psi% \rangle,\quad\frac{\partial\langle\psi\rvert}{\partial t}=\frac{i}{\hbar}% \langle\psi\rvert\hat{H}.

(2.149)

So we have

$\displaystyle\frac{\operatorname{\mathrm{d}}\!}{\operatorname{\mathrm{d}}\!t}% \langle\hat{O}\rangle$	$\displaystyle=\frac{i}{\hbar}\langle\psi\rvert\hat{H}\hat{O}\lvert\psi\rangle-% \frac{i}{\hbar}\langle\psi\rvert\hat{O}\hat{H}\lvert\psi\rangle$	(2.150)
	$\displaystyle=\frac{i}{\hbar}\langle\psi\mathclose{}\|\mathopen{}[\hat{H},\hat{% O}]\mathclose{}\|\mathopen{}\psi\rangle$	(2.151)
	$\displaystyle=\frac{i}{\hbar}\langle[\hat{H},\hat{O}]\rangle.$	(2.152)

If the commutator $[\hat{H},\hat{O}]=0$ , then the expectation value is zero. This tells us that if $\hat{O}$ does not explicitly depend on time and commutes with the Hamiltonian, its expectation value will be constant over time. This is the quantum analogue of a conserved quantity.

The result above can be applied to position and momentum to show that they actually obey Newton’s laws of motion on average. Specifically, we have

	$\displaystyle m\frac{\operatorname{\mathrm{d}}\!\langle\hat{x}\rangle}{% \operatorname{\mathrm{d}}\!t}$	$\displaystyle=\langle\hat{p}\rangle$		(2.153)
	$\displaystyle\frac{\operatorname{\mathrm{d}}\!\langle\hat{p}\rangle}{% \operatorname{\mathrm{d}}\!t}$	$\displaystyle=-\left\langle\frac{\operatorname{\mathrm{d}}\!V(x)}{% \operatorname{\mathrm{d}}\!x}\right\rangle.$		(2.154)

These two results are known as the Ehrenfest theorem.