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Suppose we have a potential barrier situation, that is $V(x)$ is zero everywhere except on the interval $[-a,a]$, where it is equal to some $V_0 > 0$. Introduce some Gaussian shaped wave packet to the left of the barrier, moving right.

What is the energy of the packet (i.e. of the system described by this wavefunction) at each instant of time?

Well, the wavefunction $\psi(x,t)$ is not an energy eigenstate, so the question is asking about the expected value of the energy, I suppose. Does that just mean carrying out the calculation

$$\langle E(t) \rangle = \int_{-\infty}^\infty dx \ \langle \psi_t | x \rangle \langle x | H \psi_t \rangle \quad ?$$

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    $\begingroup$ It seems correct. $\endgroup$
    – Sofia
    Commented Dec 23, 2014 at 22:31
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    $\begingroup$ Note that $E(t)$ is constant. The wavefunction is only a Gaussian at $t=0$, after which the shape is not so easy to compute. $\endgroup$
    – Nemis L.
    Commented Dec 23, 2014 at 22:38
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    $\begingroup$ The expected value of kinetic energy should be easy and intuitive. The expected value of $V(x)$ is harder, and will be expressible in terms of the error function. $\endgroup$
    – Nemis L.
    Commented Dec 23, 2014 at 22:40
  • $\begingroup$ The way I imagined this, it is still more or less a gaussian until it collides with the barrier, then some weird stuff happens for a while, and then its two almost-gaussians travelling in opposite direction. $\endgroup$
    – Spine Feast
    Commented Dec 23, 2014 at 22:41
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    $\begingroup$ Yes, this is correct. They are not perfect Gaussians, though, as far as I know. $\endgroup$
    – Nemis L.
    Commented Dec 23, 2014 at 22:42

1 Answer 1

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As Nemis L. pointed out, the expectation value $\langle H\rangle$ is constant, because of Ehrenfest's theorem: $$\frac{d}{dt} \langle H \rangle = \frac{1}{i \hbar} \langle [ H,H ] \rangle = 0.$$ The other way of seeing this is that the state can be written as a superposition of orthogonal energy eigenstates.

Obligatory image: Scattering of a wave packet off of a square potential barrier

Goldberg, Schey, and Schwartz, Computer Generated Motion Pictures of One Dimensional Quantum Mechanical Transmission and Reflection Phenomena", Am. J. Phys., 35, 177 (1967).)

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  • $\begingroup$ But every state can be written as a superposition of the orthogonal energy eigenstates (via Fourier transform), right? $\endgroup$
    – Spine Feast
    Commented Dec 24, 2014 at 14:46
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    $\begingroup$ Yes. It is only a reflection of conservation of energy, not something peculiar to Gaussian states. $\endgroup$
    – lionelbrits
    Commented Dec 24, 2014 at 17:31
  • $\begingroup$ I wonder, how could energy dissipation be introduced here? Is it the same as dispersion, the word I hear all the time in dispersion relation ? $\endgroup$
    – Spine Feast
    Commented Dec 24, 2014 at 23:08
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    $\begingroup$ Dispersion refers to different momenta traveling at different speeds, so that the wave-packet spreads out. In order to model energy dissipation, you would have to model the potential barrier as having it's own dynamics, which can absorb energy. An example is the Caldeira-Leggett model. $\endgroup$
    – lionelbrits
    Commented Dec 24, 2014 at 23:48
  • $\begingroup$ Thanks a lot! I always wondered how energy conservation was broken, mathmetiacially! $\endgroup$
    – Spine Feast
    Commented Dec 25, 2014 at 1:53

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