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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 Dec 23 '14 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. Dec 23 '14 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. Dec 23 '14 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 Dec 23 '14 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. Dec 23 '14 at 22:42
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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 Dec 24 '14 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 Dec 24 '14 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 Dec 24 '14 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 Dec 24 '14 at 23:48
  • $\begingroup$ Thanks a lot! I always wondered how energy conservation was broken, mathmetiacially! $\endgroup$ – Spine Feast Dec 25 '14 at 1:53

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