# Problem involving Dirac notation and Ehrenfest's theorem

Ehrenfest's theorem in one guise says (omitting hats for vectors) for suitable operators A

$$\frac{\mathrm d}{\mathrm dt}\langle A\rangle~=~\bigg\langle\frac{\partial A}{\partial t}\bigg\rangle+\bigg\langle\frac{[A,H]}{i\hbar}\bigg\rangle\tag{1}$$

and an "immediate" consequence is that, inserting the position operator x for A in the theorem, using P for the momentum operator,

$$\frac{\mathrm d}{\mathrm dt}\langle x\rangle~=~\bigg\langle\frac{P}{m}\bigg\rangle.\tag{1a}$$

Ignoring the left side of (1) I reasoned (omitting test function in expectation values)

\begin{align} \bigg\langle\frac{\partial x}{\partial t}\bigg\rangle + \bigg\langle\frac{[x,H]}{i\hbar}\bigg\rangle &=~\bigg\langle\frac{\partial x}{\partial t}\bigg\rangle + \bigg\langle\frac{xH-Hx}{i\hbar}\bigg\rangle \\ &=~\bigg\langle\frac{\partial x}{\partial t}\bigg\rangle + \bigg\langle\frac{xH}{i\hbar}\bigg\rangle~-~\bigg\langle\frac{Hx}{i\hbar}\bigg\rangle \\ &=~\bigg\langle\frac{\partial x}{\partial t}\bigg\rangle + \bigg\langle\frac{xH}{i\hbar}\bigg\rangle~-~\bigg\langle\frac{\partial x}{\partial t}\bigg\rangle \end{align} and so

$$\frac{\mathrm d}{\mathrm dt}\langle x\rangle~=~ \bigg\langle\frac{xH}{i\hbar}\bigg\rangle,\tag{2}$$

which I can convince myself is true.

On the other hand, using the first relation in this problem set (PDF):

$$[x,H]=\frac{i\hbar}{m}P,$$ we get immediately that

$$\frac{\mathrm d}{\mathrm dt}\langle x\rangle~=~\bigg\langle\frac{\partial x}{\partial t}\bigg\rangle+\bigg\langle\frac{[x,H]}{i\hbar}\bigg\rangle$$

$$\frac{\mathrm d}{\mathrm dt}\langle x\rangle~=~\bigg\langle\frac{\partial x}{\partial t}\bigg\rangle+\bigg\langle\frac{P}{m}\bigg\rangle\tag{3}$$

but then from (1a) that $\bigg\langle\partial x/\partial t\bigg\rangle~=0,$ which bothers me a little but doesn't seem to contradict anything above directly.

My question is whether the conclusions (2), (3) in these two calculations are right and if not where I went astray.

• Perhaps it would have been clearer if you did keep the operator hats on. $\partial \hat{x} / \partial t \equiv \hat{0}$ because the operator $\hat{x}$ has no explicit time dependence. Commented Jun 19, 2018 at 11:00
• @jacob1729: I will go back and edit if it's too confusing. x is the operator throughout. Commented Jun 19, 2018 at 11:04
• Note that you can use \langle and \rangle to get better looking bras and kets, instead of a bunch of less-than & greater-than symbols floating about. Commented Jun 19, 2018 at 11:06
• @daniel My point was more that maybe its confusing you. $\hat{x}$ is a constant operator, not a dynamical variable (its not the position of anything) and so it doesn't have a partial time derivative. Commented Jun 19, 2018 at 11:12
• I've edited the question adding right and left angles au lieu of greater than and smaller than signs. Commented Jun 19, 2018 at 12:33

First, notice that the term: $$\bigg\langle\frac{\partial x}{\partial t} \bigg \rangle$$ Equals zero in the Schrödinger picture (which is the one you seem to be working with) because the operators in the Schrödinger picture are constant. That is, they don't depend explicitly on time. What changes in time is their expected value, and this is represented by: $$\frac{d \langle x \rangle}{dt}$$ So one of your questions is answered (I believe).
Now, notice that: $$[x,H] = \frac{1}{2m}\big(xp^2 - p^2x\big)$$ But $[x,p]=i\hbar$ so $$\frac{1}{2m}\big(xp^2 - p^2x\big) = \frac{1}{2m}\big(xpp - ppx\big) = \frac{1}{2m}\big((i\hbar+px)p - p(xp-i\hbar)\big)$$ And so $$[x,H] = \frac{1}{2m}\big(i\hbar p + pxp - pxp + i\hbar p\big) = \frac{i\hbar p}{m}$$ It doesn't make much sense saying that $$\bigg\langle\frac{Hx}{i\hbar}\bigg\rangle = \bigg\langle\frac{\partial x}{\partial t}\bigg\rangle$$ As you seem to have done.
• The third full line is $\frac{d}{dt}<x>=<\partial x/\partial t>-<\partial x/\partial t>+<xH/ih> \implies \frac{d}{dt}<x> =~<xH/ih>$. If it's right, why doesn't the 4th line follow? Commented Jun 19, 2018 at 13:43
• That’s not right. $\bigg\langle \frac{Hx}{i\hbar \bigg\rangle$ is not equal to the expectation value of the derivative of x wrt time. Commented Jun 19, 2018 at 15:20