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flyusx
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- 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 Schrodinger 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.
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 Schrodinger 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.
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