Solving the Schrödinger Equation for a Step Potential

[laharl88]

Hi everybody! I'm studying the simple case of the solution of the Schrödinger equation for a step potential "[URL .[/URL] As my professor states , the transmission coefficient is 0 when the energy of the particle is E<V.
I really don't get how this result is not a contradiction with the fact that the wave function is not 0 for x>0

 

[elibj123]

That's confusing for me too, because your professor's statement is true only classically. Perhaps he was talking about the classic case.

Anyway, quantum mechanics (that is, Schroedinger's Equation) shows that a particle can exist in a region where his energy is lower than the potential energy governing the region (although the wave-function is exponentially decaying), and it even can penetrate a finite potential step, which is known as the Tunnel Effect.

 

[DrDu]

The transmitted wavefunction does vanish at infinity, doesn't it?
So there is no free outgoing wave.

 

[SpectraCat]

That's confusing for me too, because your professor's statement is true only classically. Perhaps he was talking about the classic case.

Anyway, quantum mechanics (that is, Schroedinger's Equation) shows that a particle can exist in a region where his energy is lower than the potential energy governing the region (although the wave-function is exponentially decaying), and it even can penetrate a finite potential step, which is known as the Tunnel Effect.

Not precisely .. for a step function the correct term in barrier penetration; tunneling requires that there be a classically allowed region on the other side of the potential barrier.

So, the transmission coefficient is in fact 0 for E<V, in the case of a step function. However, the phenomenon of barrier penetration means that the wavefunction is non-zero (and hence the probability density is finite) inside the classically forbidden region (in the case of a finite potential). There is no contradiction because the wavefunction inside the classically forbidden region decays exponentially, so in the case of a step function, it must go to zero as x goes to infinity, as DrDu mentioned.

For a step function, the real weirdness happens when E>V, because then you get non-classical reflection. That is, for most values of E/V, the reflection coefficient is non-zero, but in the classical case there is no way it should be able to reflect from the barrier.

 

[laharl88]

There is no contradiction because the wavefunction inside the classically forbidden region decays exponentially, so in the case of a step function, it must go to zero as x goes to infinity, as DrDu mentioned.

I know that the wave function goes to zero, but if I do the integral of |psi|^2 between a and b for example, where a and b belong to the region in which the potential is V, then i would have a non-zero probability to find the particle in that region of space!

 

[SpectraCat]

I know that the wave function goes to zero, but if I do the integral of |psi|^2 between a and b for example, where a and b belong to the region in which the potential is V, then i would have a non-zero probability to find the particle in that region of space!

But transmission is defined in terms of classical propagation beyond the barrier. The components of the probability density inside classically forbidden regions (CFR's) are not included in calculations of transmission and reflection coefficients.

Also, although the probability to find the particle inside the barrier is non-zero, you will find that it is impossible to carry out an experiment where one can be certain that the particle was observed while inside a CFR. I had an extensive discussion on PF about this a while back that you may find interesting: https://www.physicsforums.com/showthread.php?t=372423"

 

[laharl88]

Well, i read the other topic but i think that didn't help much.

But transmission is defined in terms of classical propagation beyond the barrier. The components of the probability density inside classically forbidden regions (CFR's) are not included in calculations of transmission and reflection coefficients.

If that's the case then why do we define these quantities in quantum mechanics?

Also, although the probability to find the particle inside the barrier is non-zero, you will find that it is impossible to carry out an experiment where one can be certain that the particle was observed while inside a CFR. I had an extensive discussion on PF about this a while back that you may find interesting: https://www.physicsforums.com/showthread.php?t=372423"

As you say in the other topic (if I understood correctly), post number 9, an experiment would perturbate the particle and could give it such energy to make the previously CFR into a CAR...
But that doesn't make me understand: apart from experiments, how do i explain theoretically the non zero probability of finding the particle in CFR?
You also said that in order to exist in CFR, the particle would have an imaginary momentum, which clearly sounds absurd, so my question is: if it's not possible to find the particle in CFR, why in the first place do i define the wave function in that region?

 

[SpectraCat]

Well, i read the other topic but i think that didn't help much.
If that's the case then why do we define these quantities in quantum mechanics?
As you say in the other topic (if I understood correctly), post number 9, an experiment would perturbate the particle and could give it such energy to make the previously CFR into a CAR...
But that doesn't make me understand: apart from experiments, how do i explain theoretically the non zero probability of finding the particle in CFR?
You also said that in order to exist in CFR, the particle would have an imaginary momentum, which clearly sounds absurd, so my question is: if it's not possible to find the particle in CFR, why in the first place do i define the wave function in that region?

This is mostly a problem of language and specificity I think. I try to be clear and consistent all the time, but perhaps I don't always succeed.

First: Transmission and reflection are classical concepts ... quantum mechanics is completely consistent with all classical results ... therefore transmission and reflection coefficients are reasonable things to compute for a QM system. The reason that is useful in the case of these 1-D "toy" problems, is because it helps highlight some unexpected results, namely non-classical reflection and tunneling (in the case of a finite width barrier).

Second: It is a postulate of quantum mechanics that the wavefunction of a system yields all the information about that system. Therefore, assuming your Hamiltonian gives a complete description of the system (which is trivial in these simple problems), whatever wavefunction you derive from the Schrödinger equation must be the correct one. If it shows non-zero values inside a CFR, then so be it; that means there is non-zero probability density inside the CFR, and we must make whatever sense of it we can. The imaginary momentum is an unfortunate example of this ... if we use our classically-based logic to analyze that little bitty portion of the wavefunction that exists inside the barrier, we come up with nonsense. That is because it is the *entire* wavefunction that describes the system, and any measurement we make reflects a measurement of the *entire* wavefunction ... we can't measure just part of it.

Third: One has to be careful with wording when making statements like, "it is impossible to find the particle in the CFR" ... that is why I worded my response the way I did. You are free to measure the position of the particle however you see fit. However, if you happen to measure a value that corresponds to the particle being inside the CFR for, say, the ground state, you will always find that you can no longer be certain that the particle was in that state when the measurement occurred. For example, you will find that your measurement technique allows for the possibility of adding enough energy to the system to excite it to another state where the position you measured *doesn't* correspond to a CFR. This is a consequence of the HUP, as I said in that other thread.

Hope this helps.

 

[DrDu]

I think it is misleading to say that transmission and reflection are classical concepts, at least when referring to a particle. These are concepts which come mainly from experiments with electromagnetic waves and their "classical" behaviour is quite similar to the qm behaviour of particles. For em waves one also observes that they can enter into classically forbidden regions where they decay exponentially (google e.g. for the Goos-Haenchen effect) and one observes also quite trivially reflection "over the barrier", like the reflection of light from a glass plate.
The definition of reflected and transmitted wave really refers only to free waves which can be observed at a large distance from the step. This is quite natural in the language of scattering theory where one considers only how an asymptotically free incoming wave is scattered into asymptotically free outgoing waves.

 

[SpectraCat]

I think it is misleading to say that transmission and reflection are classical concepts, at least when referring to a particle. These are concepts which come mainly from experiments with electromagnetic waves and their "classical" behaviour is quite similar to the qm behaviour of particles. For em waves one also observes that they can enter into classically forbidden regions where they decay exponentially (google e.g. for the Goos-Haenchen effect) and one observes also quite trivially reflection "over the barrier", like the reflection of light from a glass plate.
The definition of reflected and transmitted wave really refers only to free waves which can be observed at a large distance from the step. This is quite natural in the language of scattering theory where one considers only how an asymptotically free incoming wave is scattered into asymptotically free outgoing waves.

Well, I guess I see what you are saying, but I don't understand how any of that is significantly different from what I was saying. All the phenomena you describe are for classical waves, and I don't agree the Goos Haenchen effect involves a "classically forbidden region". Evanescent waves are a classical phenomenon, and the "barriers" in such cases are described in terms of their indicies of refraction. IMO, it is therefore quite surprising to find similar behavior for massive particles, where the analog to the index of refraction is related to the total energy of the system.

Since we are dealing with massive particles here in the current context, I think it is more natural to think about transmission and reflection as defined in scattering theory, and that is what I was addressing in my post. I also don't agree that T and R only make sense for free waves .. one can define them equally well for a propagating wave-packet in the time-dependent case, and they mean the same thing.

 

[DrDu]

Dear spectracat,
I agree with you 100%. I just wanted to point out that things including even evanescent waves -which are the analogue of barrier penetration of massive particles- are described in classical EM, and therefore all this language - transmission and reflection - is not something which has been invented in quantum mechanics. This refers mainly to post #7.
I also agree with you (and wrote so) that these definitions are best understood in the context of scattering theory.