E<V Potential Step Confusion

[flyusx]

Homework Statement

For the potential outlined below, in the case where $E<V$, the transmission coefficient is zero for the region $x\geq0$ but the wave function is also non-zero suggesting the particle can be found in the this region?

Relevant Equations

$$\Psi(x,t)_{x<0}=A\exp\left(i\left(k_{1}x-\omega t\right)\right)+B\exp\left(-i\left(k_{1}x+\omega t\right)\right)$$
$$\Psi(x,t)_{x\geq0}=C\exp\left(-k_{2}'x-\omega t\right)$$
$$R=\frac{\vert B\vert^{2}}{\vert A\vert^{2}}$$

This isn't really homework but rather me working through my quantum textbook and coming across something I don't understand. Consider the potential function $$V(x)=\begin{cases}0&x<0\\V&x\geq0\end{cases}$$ where $V$ is a constant. If the energy of the incident particle $E$ is less than $V$, the reflection coefficient is 1 and the transmission coefficient is zero. I have no problem with the way these are calculated.

The wave function is, however, able to penetrate the barrier. It takes the form $$\Psi(x,t)=C\exp\left(-k_{2}'x-\omega t\right)$$ where $$k_{2}'=\frac{\sqrt{2m(V-E)}}{\hbar}$$ as is found by solving the Schrödinger equation for the region $x\geq0$. I have no problem with this calculation either.

My confusion arises when I try to interpret these relations. In the case where $E>V$, it is my understanding that the transmission and reflection coefficients are used to determine the probability of a particle being found in $x<0$ or $x\geq0$ (perhaps because the wave functions are non-normalisable?). However, since a wave function is associated with probability density and the wave function is non-zero for $x\geq0$ in the $E<V$ case, how does that not conflict with the transmission coefficient being zero? Maybe because transmission coefficients only make sense for plane waves?

Looking ahead in my book (Zettili Chapter 4.5), I see the next section is on the potential barrier which absolutely can involve quantum tunnelling. Something in my understanding must be wrong and I'd like to figure that out before I go further.

 

[anuttarasammyak]

In that case k is pure imaginary so wave function does not oscillate but decays exponentially.
cf https://en.wikipedia.org/wiki/Finite_potential_well

 

[flyusx]

In that case k is pure imaginary so wave function does not oscillate but decays exponentially.

Wouldn't exponential decay for $x\geq0$ still suggest $\vert\psi\vert^{2}\neq0$ and hence the probability density is non-zero for this region (the particle can tunnel into the potential barrier)?
If so, how would one go about calculating this probability for the transmission coefficient is zero and the wave functions are non-normalisable.

 

[vela]

The coefficients compare probability flux. When the wave function is real (not oscillatory), there is no flow of probability.

 

[flyusx]

The coefficients compare probability flux. When the wave function is real (not oscillatory), there is no flow of probability.

Looking from that perspective, it makes sense how why the transmission coefficient is zero in this case; the wave function (and hence its probability density) is static for $x\geq0$.

On the wikipedia page for transmission coefficients, I read the following:

"In non-relativistic quantum mechanics, the transmission coefficient and related reflection coefficient are used to describe the behavior of waves incident on a barrier. The transmission coefficient represents the probability flux of the transmitted wave relative to that of the incident wave. This coefficient is often used to describe the probability of a particle tunneling through a barrier."

What I now don't understand is that if the transmission coefficient is zero, wouldn't this mean the probability of an incident particle tunneling through the barrier for E<V is zero? Yet the wave function for $x\geq0$ is non-zero so there would be an associated non-zero probability density in this region. I must be understanding something wrong.

Thanks in advance!

 

[anuttarasammyak]

Looking ahead in my book (Zettili Chapter 4.5), I see the next section is on the potential barrier which absolutely can involve quantum tunnelling. Something in my understanding must be wrong and I'd like to figure that out before I go further.

Why don’t you proceed to the next section and revisit your problem of a infinite long tunnel? Good organized text books are much reliable than Wikipedia.

 

[flyusx]

Why don’t you proceed to the next section and revisit your problem of a infinite long tunnel? Good organized text books are much reliable than Wikipedia.

Reading a few (or quite a few) more posts in the past few hours has given me the impression that the wave could transmit through the barrier, 'bleed into' it and then get reflected. All of the wave will eventually get reflected so $R=1$ and $T=0$. One could hypothetically measure the particle in $x\geq0$ but that's irrespective of $R$ and $T$.

I'll continue onto the potential barrier and wells.

 

[hutchphd]

Reading a few (or quite a few) more posts in the past few hours has given me the impression that the wave could transmit through the barrier, 'bleed into' it and then get reflected. All of the wave will eventually get reflected so $R=1$ and $T=0$. One could hypothetically measure the particle in $x\geq0$ but that's irrespective of $R$ and $T$.

I'll continue onto the potential barrier and wells.

Note also that T and R are not necessarilly real numbers. In this case the phase R will change with penetration even though |R|=1. Note then that a packet of waves will change shape and suffer a time delay upon reflection depending upon penetration depth. Lotsa physics here.