Can Bound States Leak and How Do Scattering States Behave?

[WWCY]

Hi all, I'd like some assistance regarding some issues I have understanding such states. (Referencing Griffiths' QM)

1) Meaning of Bound and Scattering States.

The bound states I have studied thus far are limited to the infinite square well and the quantum harmonic oscillator. In the case of a harmonic oscillator, a particle is able to "leak" out of its potential well.

a) Would I be right in saying that even bound states can experience this "leaking" into classically forbidden zones (finite potentials only), but is limited to the vicinity of its "well"?

b) On the other hand, is it right to say that a scattering state can not only be found in classically forbidden zones, but is itself a also traveling wave (unlike the bound state)? I.e it is not bound by any sort of "well" and is free to move about

2) Delta-function potential.

If we were to work out the (bound-state) wave-function for a potential $-\alpha \delta (x)$, this would yield a wave-function peaked at x = 0. However, there is a non-zero probability that we can find said particle outside the delta-function well.

a) Is this infinitely deep well not analogous to an infinite square well, except that its depth is from 0 to $-\infty$ rather than 0 to $\infty$? Why would we still expect that the particle can be found outside of the vicinity of x = 0?

b) Is the Delta-function potential considered to be an even, or odd function? Scouring the internet seems to bring up conflicting opinions.

3) Even potentials and their solutions.

a) It was stated in the book that an even potential gives rise to either even or odd solutions to the Schrödinger equation. Say I obtain the even and odd solutions separately, in some domain of x. Do I then sum the even and odd solutions in their respective domains to get the actual wavefunction? I.e is the actual wavefunction for an even potential a sum of even and odd solutions?

Thanks very much in advance!

PS: Would it be possible to discuss these in a not-so-technical manner for now?

 

[blue_leaf77]

a) Would I be right in saying that even bound states can experience this "leaking" into classically forbidden zones (finite potentials only), but is limited to the vicinity of its "well"?

Yes, unless the well is of infinite height. However, even for infinitely growing potential like harmonic oscillator potential, the eigenfunctions have nonzero portion in the forbidden region.

b) On the other hand, is it right to say that a scattering state can not only be found in classically forbidden zones, but is itself a also traveling wave (unlike the bound state)? I.e it is not bound by any sort of "well" and is free to move about

For scattering state, the classically forbidden region is not applicable since the energy of such state must always be more positive than the potential anywhere. Generally speaking, scattering state is an energy eigenfunction in the continuum spectrum and has nonzero probability current, i.e. it's traveling. This type of eigenfunction is an approximate eigenfunction of momentum, i.e. it has an almost definite momentum. But sometimes the term scattering state is also used to refer to an energy eigenfunction in the continuum but has its probability current vanishing. Usually in this case, the eigenfunction is also an eigenfunction of total angular momentum. On the other hand, its momentum distribution has a large uncertainty.

a) It was stated in the book that an even potential gives rise to either even or odd solutions to the Schrödinger equation. Say I obtain the even and odd solutions separately, in some domain of x. Do I then sum the even and odd solutions in their respective domains to get the actual wavefunction? I.e is the actual wavefunction for an even potential a sum of even and odd solutions?

No, the even and odd solutions are the eigenfunctions of an even potential separately. A wavefunction formed by a sum of even and odd solutions is no longer an energy eigenfunction since it does not have a definite parity/even(odd)ness.

 

[WWCY]

Hi, thanks for the response!

For scattering state, the classically forbidden region is not applicable since the energy of such state must always be more positive than the potential anywhere. Generally speaking, scattering state is an energy eigenfunction in the continuum spectrum and has nonzero probability current, i.e. it's traveling. This type of eigenfunction is an approximate eigenfunction of momentum, i.e. it has an almost definite momentum.

Is this in reference to the wave-packet equation for the free particle? Also, how do we tell if momentum is almost definite? The text seems to suggest that the certainty with which we can know a particle's momentum is determined by the spread of $k$ in the Fourier transform term in the wave-packet equation.

 

[blue_leaf77]

Is this in reference to the wave-packet equation for the free particle?

No, it applies to general (real) potential.

Also, how do we tell if momentum is almost definite? The text seems to suggest that the certainty with which we can know a particle's momentum is determined by the spread of kk in the Fourier transform term in the wave-packet equation.

Yes, as your text suggests, a state with almost definite momentum has a narrow width of the Fourier transform of the wavefunction in position space.