(a) Write down the wave functions for the three regions of...

(a) Write down the wave functions for the three regions of the potential energy barrier (Figure 5.25 ) for $E<U_{0}$. You will need six coefficients in all. Use complex exponential notation. ( $b$ ) Use the boundary conditions at $x=0$ and at $x=L$ to find four relationships among the six coefficients. (Do not try to solve these relationships.) (c) Suppose particles are incident on the barrier from the left. Which coefficient should be set to zero? Why?

Key Concepts

Quantum Tunneling

Quantum tunneling refers to the phenomenon in quantum mechanics where particles have a probability to cross a potential barrier even when their energy is less than the barrier height. This is a direct consequence of the wave-like nature of particles, as described by the Schrödinger equation, allowing them to penetrate into classically forbidden regions.

Schrödinger Equation and Wave Function Representation

The Schrödinger equation is the fundamental equation governing quantum systems. For problems involving a potential barrier, its solutions are represented as wave functions in different regions. These wave functions are typically expressed in terms of complex exponentials, which capture both oscillatory behavior in regions where the particle's energy exceeds the potential and exponential decay in regions where the energy is lower than the barrier height.

Boundary Conditions

Boundary conditions are essential in ensuring that the physical solutions to the Schrödinger equation are consistent and continuous across different regions. They require that both the wave function and its derivative remain continuous at the interfaces between regions with different potentials. These conditions give rise to relationships among the amplitude coefficients in each region, linking the incident, reflected, and transmitted wave components.

Setting Coefficients Based on Physical Conditions

In scenarios where particles are incident from one side only, physical intuition dictates that there should be no incoming wave from the opposite side. This requirement is implemented by setting the coefficient corresponding to that non-existent incoming wave to zero. This simplifies the mathematical analysis and accurately reflects the one-sided incidence of particles.

Transcript

00:01 In this problem, we have a potential barrier, and we're also given that the height of this barrier is greater than the total energy of the wave.

00:08 And we need to write down the wave function in the three regions.

00:12 So i'll call these three regions region one up to x equals zero.

00:18 Region two is this region, and notice this is a classically forbidden region, so we're just going to have tunneling in here.

00:24 And then finally, this will be region three.

00:27 So to write down the wave functions, it's just going to be the general.

00:32 Forms of a free particle.

00:34 So we're going to have a term that represents an incoming wave traveling to the right, and it's going to have wave vector k -0.

00:43 And then we have another term representing the reflected wave traveling to the left.

00:53 And these are oscillatory because the kinetic energy is positive as it should be in this region.

00:59 And i might as well do the same for the third region.

01:04 I'll call these coefficients f and g.

01:07 So this is representing a traveling wave to the right, and this is representing a traveling wave to the left.

01:15 And notice that the wave vector is k -0 in both cases, because the potential energy is the same in both cases.

01:23 Finally, this middle region, we're not going to have oscillatory exponentials.

01:27 We're going to have rising and decaying exponentials.

01:32 So this will be c, some new constant, times k1, there's no i.

01:40 This is just a pure rising exponential.

01:43 And another one, which is a falling exponential, k1x.

01:49 This k1 is different from k0.

01:52 And these k zeros and k -1s are related to the energy like this...

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