Question

In this problem, you will calculate the transmission probability through the barrier illustrated in Figure 16.10 We first go through the mathematics leading to the solution. You will then carry out further calculations. The domain in which the calculation is carried out is divided into three regions for which the potentials are $$\begin{array}{lll}V(x)=0 & \text { for } x \leq 0 & \text { Region I } \\\ V(x)=V_{0} & \text { for } 0

Short answer

The transmission probability is given by: $$ \frac{F}{A} = \sqrt{\frac{1}{1+\left[\left(k^{2}+\kappa^{2}\right)^{2} \sinh ^{2}(\kappa a)\right] / 4(\kappa k)^{2}}} $$ By solving this expression using numerical methods, you can determine the energy values at which the tunneling probabilities are 0.1 and 0.02, and the barrier width at which the transmission probability is 0.2.

Step-by-step solution

Establish Conditions

We will use the wave function expressions to find the conditions for the coefficients A, B, C, D and F, which are: $$ A+B=C+D $$ $$ Ce^{-\kappa a} + De^{\kappa a} = Fe^{+ik a} $$ $$ A - B = -\frac{i \kappa}{k}(-C+D) - Ce^{-\kappa a} + De^{\kappa a} = \frac{i k}{\kappa} Fe^{+ik a} $$

Express A and B in terms of C, D, and F

Now, we can solve for D and C in terms of F using the equations derived in Step 1 to get: $$ D=\frac{i k e^{+i k a}+\kappa e^{+i k a}}{2 \kappa e^{+\kappa a}} F $$ $$ C=\frac{-i k e^{+i k a}+\kappa e^{+i k a}}{2 \kappa e^{-\kappa a}} F $$ And substituting D and C, we find: $$ A=\frac{(i k-\kappa) C+(i k+\kappa) D}{2 i k} $$

Relate A and F

Now, substitute the results for C and D in terms of F into the equation derived in Step 2 to get: $$ 2ik A = \frac{e^{+ik a}}{2 \kappa}\left[(i k-\kappa)(-i k+\kappa) e^{+\kappa a}+(i k+\kappa)(i k+\kappa) e^{-\kappa a}\right] F $$

Derive Transmission Probability

Using hyperbolic trigonometric functions and the relationship between A and F, derive the expression for the transmission probability: $$ \frac{F}{A} = \sqrt{\frac{1}{1+\left[\left(k^{2}+\kappa^{2}\right)^{2} \sinh ^{2}(\kappa a)\right] / 4(\kappa k)^{2}}} $$ Then, the part e and f can be done with the help of numerical analysis and plotting software, such as Python or Matlab, to find the energy values and the barrier width at which the transmission probabilities are 0.1, 0.02, and 0.2, respectively.

Key concepts

Quantum Tunneling

Quantum tunneling is a fascinating quantum mechanical phenomenon that allows particles such as electrons to pass through a potential barrier, even when they do not possess enough energy to overcome it by classical means. This concept is comparable to a ball rolling up a hill but, instead of stopping and rolling back down because it can't go over the top, it somehow magically appears on the other side of the hill without having climbed it. This counterintuitive behavior arises from the wave-like properties of particles at the quantum level.

In classical mechanics, if an object lacks the energy to surmount a barrier, the situation is a dead end. However, quantum mechanics introduces the probability of tunneling, where a particle can penetrate and travel through a barrier to emerge on the other side. This has enormous implications in fields ranging from quantum computing to the inner workings of nuclear fusion in stars.

Transmission Probability

Transmission probability quantifies the likelihood of a particle tunneling through a potential barrier. Within the context of quantum mechanics, it is given by the square of the absolute value of the ratio of the transmitted wave amplitude to the incident wave amplitude (|F/A|^2). This value is derived using the Schrödinger equation, a fundamental equation in quantum mechanics, which describes how the quantum state of a physical system changes over time.

The steps laid out in the exercise provide a way to mathematically calculate this probability by connecting the coefficients of the wave functions across different regions. These procedures ensure that despite the seemingly bewildering concept of tunneling, there is a concrete method to predict its occurrence. The exercise further demonstrates how this probability varies with certain parameters, such as the energy of the particle or the width and height of the barrier it faces.

Potential Barrier

In quantum mechanics, a potential barrier is an energy threshold that a particle would typically need to have enough energy to cross. Imagine confronting a wall or hill; if you lack the energy (or, say, the ability to jump high enough), you cannot pass it. Quantum mechanically, this barrier is often depicted as a step-function in potential, where the potential energy suddenly changes value over a small distance, as illustrated in the exercise. It's like a hurdle in track and field; to get over it, you need a certain amount of kinetic energy.

In the exercise, an electron confronts a barrier with potential energy V_0, but its quantum mechanical nature opens up the possibility of tunneling through this barrier, instead of being solely reflected back. The fascinating aspect is that this tunneling can occur even when the energy of the electron is less than the energy of the barrier (E < V_0).

Wave Function Continuity

The wave function in quantum mechanics describes the state of a quantum system, encoding all the information about the system's properties, and its behavior is governed by the Schrödinger equation. A fundamental property of the wave function is its continuity; it must be smooth and unbroken across space. This requirement leads to constraints on the behavior of wave functions when encountering potential barriers.

As manifested in the exercise, ensuring wave function continuity at the junctions of different potential regions (e.g., at x=0 and x=a) helps determine the coefficients of the wave functions in each region. The continuity of both the wave functions and their first derivatives at the boundaries of the regions is pivotal in correctly solving quantum mechanical problems and is an inherent part of the conditions set to find the transmission probability through a potential barrier.