I had an assignment question in which I was asked to calculate the expectation value of energy, $\langle E\rangle (t),$ and in the solution to it, the following was stated:

\begin{align*} \langle E\rangle (t) &=\langle \Psi(x, t)|H| \Psi(x, t) \rangle \\ &= \langle \Psi(x, t)|e^{-iHt} |\Psi(x, t)\rangle \\ &= \langle \Psi(x, 0)| H |\Psi(x, 0)\rangle \end{align*}

I don't how the first and third lines are exactly the same, just the difference of the $t$ being replaced by $0$ in the last line? Can somebody please help me understand this?

  • $\begingroup$ How strange - the \bk MathJax tag renders correctly in the question preview, but not when it is officially displayed on the question page. Some kind of bug, or maybe it's just my browser? $\endgroup$
    – Brionius
    Dec 10, 2015 at 12:33
  • $\begingroup$ I fixed it, by writing out the braket notation by hand. And I see the problem: The definition has to be at the top of the article. But my experience is that the support for \newcommand and \def is flakey (I observed inconsistent behaviour as well). $\endgroup$ Dec 10, 2015 at 12:34
  • $\begingroup$ Now for the content: There must be something wrong in the second line, note that $\lvert \Psi(t) \rangle = e^{iHt}\lvert \Psi(0) \rangle$ by definition. Then the correct solution is obvious. $\endgroup$ Dec 10, 2015 at 12:37
  • $\begingroup$ @SebastianRiese you beat me to it. Still feels like buggy behaviour to me. See this meta thread for details. $\endgroup$ Dec 10, 2015 at 12:46

1 Answer 1


If this is indeed how your text is stated, it is incorrect. I would phrase this as \begin{align*} \langle E\rangle (t) &=\langle \Psi(x, t)|H| \Psi(x, t) \rangle \\ &= \langle \Psi(x, 0)|e^{+iHt}He^{-iHt} |\Psi(x, 0)\rangle \\ &= \langle \Psi(x, 0)|He^{+iHt}e^{-iHt} |\Psi(x, 0)\rangle \\ &= \langle \Psi(x, 0)| H |\Psi(x, 0)\rangle, \end{align*} where $H$ commutes with $e^{\pm iHt}$ because the latter is a function of $H$, and the two exponentials are mutual inverses. In essence, what this is saying is that $\langle E\rangle$ is conserved: it is the same no matter what time $t$ you evaluate it at. There's really not much more going on.

  • $\begingroup$ Ahh, that makes sense. Must have been a typo. Thank you so much! $\endgroup$ Dec 13, 2015 at 13:21

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