(a) Write down the wave functions for the three regions of the potential energy barrier (Figure

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

Key Concepts

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.

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.

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.

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.

Transcript

00:01 In this problem, we're asked to analyze the potential step with incoming particles on the left, having energy less than the height of the potential barrier.

00:13 We're asked to figure out, we're asking to write down the wave functions in each region, and use the boundary conditions to find the equations that relate the undetermined amplitude parameters to each other.

00:28 So let's start by the region to the left of the boundary, when x is less than the one.

00:36 And 0.

00:37 Here we have the wave function and it will have an incoming piece denoted by e to the ik0x which represents a right moving wave with wave number k -0 and it might have some reflection from the boundary which will be represented by u to the minus i k -0 x which is for a left -moving wave.

01:05 A and b are undetermined amplitude coefficients.

01:08 And k0, and there's no potential energy, is given by squared of 2m .e over hbar squared.

01:25 In region 1, where x is between x0 and l, we can still do the same thing.

01:33 Psi 1 of x is, now let's name this coefficient c.

01:38 In this case, because e is less than u0, there's not really traveling ways, but it's more like exponential decays and increases.

01:49 And in that case we just simply write e to the k1x plus d to the e minus k1x, where k1 is equal to 2m u0 minus e divided by h bar squared.

02:16 For the final region where x is larger than l, let's call that side 2 of x.

02:23 Again, we have to write down some outer coefficients.

02:26 Let's call it f.

02:28 Now this one is representing a way that is traveling to the right, and g, u to the minus ik2 of x, and this one is representing a wave traveling to left, and k2 is going to actually be equal to k0, because the energy is the same and there's the potential here.

02:50 So this again is equal to 2m over h -bar square.

02:59 Now these are wave functions, size 0 for when x is less than 0 in this region, psi 1 in the potential barrier, and si 2 in the right region.

03:14 Now, whenever there is a boundary that the wave function has two pieces on each side, we have to use continuity equations to relate the coefficients to the level letter, and continuity equations are basically for x equals 0, for the boundary at x equals 0, these relations are that the wave functions values on both sides have to be equal to one another, and the derivative rotovay function, evaluated at the boundary, have to be equal to each other on both sides as well.

04:04 Now, this one will tell us that a, e to the 0, plus b, e to the 0, ik0, ik0, ik0000, is equal to c, e to the k10 plus d, that is a minus here, sorry, e to the minus k10.

04:33 Thus, this will tell us a plus b equals c plus d.

04:41 Now let's call this the derivative expression b and evaluate that.

04:48 The derivative of size 0, we expect to x, is simply equal to ik0, a, e to the ik0 x, minus ik0, b e to the minus ik0x.

05:11 And if we evaluate this at x equals 0, we will get rid of the exponentials...

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