# Expectation of momentum in the bound state

Is it logically correct to assert that the expectation of the momentum $$\langle \hat p \rangle=0$$ for any bound state because it is bound to some finite region? What is the physical interpretation of the fact that $$\langle \hat p \rangle=0$$ in an energy eigenstate $\psi_n(x,t)$ but $$\langle \hat p \rangle\neq0$$ in some superposition state $$\psi(x,t)=c_m\psi_m(x,t)+c_n\psi_n(x,t)~?$$ Here $\psi_n(x,t)$ the eigenstates of the Hamiltonian, for example, in the problem of particle in a box (say).

• I think one can only conclude for non-relativistic, bound, eigenstate $\langle \hat{p} \rangle=0$. Since $$\langle n | p | n \rangle \propto \langle n | [H,x] | n \rangle = \langle n | Hx-xH | n \rangle = E_n ( \langle n| x | n \rangle - \langle n| x | n \rangle) =0$$, here $$H=\frac{p^2}{2m} + V$$. If we relax the state into any bound state $| \rangle$, we have $$\langle | p | \rangle \propto \langle | [H,x] | \rangle = \sum_n c^*_n E_n \langle n | x | \rangle - c_n E_n\langle |x | n \rangle \neq0$$ in general. – user26143 Feb 22 '14 at 7:52
• @user26143 Why not turn that into an answer? – user10851 Feb 22 '14 at 9:22
• My reply is a technical note. I am not sure how to answer "What is the physical interpretation of the fact that ⟨p^⟩=0 in an energy eigenstate ψn(x,t) but ⟨p^⟩≠0 in some superposition state" I guess it relates to shape of wavefunction?... Perhaps I will turn it into an answer – user26143 Feb 22 '14 at 15:45
• if the potential has symmetry along a given direction, then it follows that an eigenstate will move along that direction in the same way regardless of the orientation, hence it must cancel out to zero when averaged – lurscher Oct 22 '18 at 19:06

Is it logically correct to assert that the expectation of the momentum $\langle p \rangle=0$ for any bound state because it is bound to some finite region?

Bound state means the particles are bounded somewhere. Its wavefunction will vanish at the asymptotic limit. A bound state could be a superposition of a finite number of bound eigenstates. For instance, the superposition of the ground and first excited-state wavefunction of particle-in-box will still vanish at far limit.

I think one can only conclude for non-relativistic, bound, eigenstate (not any bound state) $\langle \hat{p} \rangle=0$. Since $$\langle n | p | n \rangle \sim \langle n | [H,x] | n \rangle = \langle n | Hx-xH | n \rangle = E_n ( \langle n| x | n \rangle - \langle n| x | n \rangle) =0$$. If we relax the state into any bound state $| \rangle$, we have $$\langle | p | \rangle \sim \langle | [H,x] | \rangle = \sum_n c^*_n E_n \langle n | x | \rangle - c_n E_n\langle |x | n \rangle \neq0$$ in general.

• Why won't the argument hold for unbounded states? – GuillemB Jan 31 '17 at 15:27


Where in (1) I integrated by parts and assumed that $\psi \to 0$ as $x \to \infty$ and in (2) I used the fact that you can always choose a bounded energy eigenstate to be real, which implies that I can take the complex conjugate at no cost. Notice that we have the following:

$$\bk{\psi}{p}{\psi} \propto - \int \mathrm{d}x \, \psi' \psi^* = \int \mathrm{d}x \, \psi' \psi^* \iff \bk{\psi}{p}{\psi} = 0$$

A hint on this could be the fact that a superposition of stationary states of different energies is NOT a stationary state, because you can not express the wave function as the product of a single time-dependent exponential tiames a spatial function.

I know this has already been answered, but I think there is a nice way to see this that hasn't been mentioned. If the state in question is a stationary state (energy eigenstate), then we know

$$H|\Psi\rangle=E|\Psi\rangle$$

which means that

$$\langle[x,H]\rangle=\langle\Psi|xH-Hx|\Psi\rangle=E\Big(\langle\Psi|x|\Psi\rangle-\langle\Psi|x|\Psi\rangle\Big)=0,$$

and since $$\hat{x}$$ has no explicit time dependence we have the simple differential equation for the $$\langle x\rangle :$$

$$\frac{d\langle x\rangle}{dt}=\frac{1}{i\hbar}\langle [x,H]\rangle =0.$$

Now recall the immensely useful commutation rule

$$[x,F(p)] = i\hbar\frac{\partial F}{\partial p}.$$

Since the potential only depends on position, it commutes with $$x$$, so the above time derivative can be written

$$0=\frac{d\langle x\rangle}{dt}=\frac{1}{i\hbar}\langle [x,H]\rangle = \frac{1}{i\hbar}\langle [x,\frac{p^2}{2m}]\rangle = \frac{1}{2m}\langle\frac{\partial}{\partial p}p^2\rangle = \frac{\langle p\rangle}{m}.$$

So we see that the expectation of potential for an energy eigenstate is zero:

$$\langle p\rangle = 0.$$