Question

Whereas an infinite well has an infinite number of bound states, a finite well does not. By relating the well height $U_0$ to the kinetic energy and the kinetic energy (through $\lambda$ ) to $n$ and $L$. show that the number of bound states is given roughly by $\sqrt{8m{L}^2{U}_0/h^2}$. (Assume that the number is Large.)

Short answer

The number of bound states in a finite potential well is given by the expression $n=\sqrt{\frac{8m{L}^2{U}_0}{h^2}}$.

Step-by-step solution

Understanding the problem

The number of bound states in a quantum well represents the number of wavefunctions that can exist within that well, and these depend on the depth and width of the well (given by $U_0$ and $L$ respectively). In a finite well, the number of states is not infinite and we must infer a relationship that represents these bound states.

Using the Schrödinger equation

First, we can use the time-independent Schrödinger equation, which states $-\frac{h^2}{8m}\frac{d^2\psi }{d{x}^2}+U_0\psi =E\psi$, where $E$ is the total energy of the system, $\psi$ is the wave function, and $U_0$ is the potential energy (well height). Here, $\lambda$ is related to kinetic energy by the expression $\lambda =\sqrt{\frac{8mE}{h^2}}$. Let's substitute this back into the Schrödinger equation.

Substituting in the equation and simplification

Substituting the expression for lambda into the Schrödinger equation gives us $-\frac{1}{{\lambda }^2}\frac{d^2\psi }{d{x}^2}+U_0\psi =E\psi$. Rearranging, we get $U_0=E+\frac{1}{{\lambda }^2}\frac{d^2\psi }{d{x}^2}$, which shows that the potential energy is partly dependent on kinetic energy and wave function.

Relating energy to well parameters

We have now a relationship of potential energy with other parameters, but this still is not our final goal. Our task is to show that the number of bound states $n$ is roughly given by $\sqrt{8m{L}^2{U}_0/h^2}$. Now, we can express kinetic energy $K=E-U_0$ in terms of well parameters $L$ and $U_0$. From the definition of kinetic energy, and using the relationship $\lambda =\sqrt{\frac{8mE}{h^2}}$, we get $K=\frac{{\lambda }^2{h}^2}{8m}-U_0$.

Expressing number of bound states

The quantization condition for the number of bound states $n$ in a well can be expressed as $n=\frac{L\lambda }{\pi }$. Substituting our expression for $\lambda$, we find $n=\frac{L\sqrt{\frac{8mE}{h^2}}}{\pi }$. Now, using our relationship for kinetic energy, the expression for the number of bound states transforms into $n=\frac{L}{\pi }\sqrt{\frac{8m(\frac{{\lambda }^2{h}^2}{8m}+U_0)}{h^2}}$, and simplifies to $n=\sqrt{\frac{8m{L}^2{U}_0}{h^2}}$.

Conclusion

We have derived the relationship given, showing that the number of bound states $n$ in a finite potential well is roughly given by $\sqrt{8m{L}^2{U}_0/h^2}$.

Key concepts

Quantum Mechanics

Quantum mechanics is the fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is a mathematical framework that allows for the calculation of the probabilities of different outcomes in measurements of quantum systems.

Unlike classical physics, which is deterministic and predicts exact outcomes, quantum mechanics is inherently probabilistic. This means that we can only make predictions about the likelihood of where a particle might be or how it might behave, not exact certainties.

At the heart of quantum mechanics is the concept of the wavefunction, often denoted as $\psi$, which encapsulates all the information about a quantum system. It is this wavefunction that is manipulated by the Schrödinger equation to predict behavior in quantum systems, such as electrons in an atom or photons of light.

Schrödinger Equation

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. For stationary problems, we often use the time-independent Schrödinger equation, which looks at the system's state at a constant energy level without considering temporal changes.

The time-independent Schrödinger equation can be expressed as $-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}+V(x)\psi =E\psi$ where $E$ is the energy of the system, $\psi$ is the wavefunction, $V(x)$ is the potential energy as a function of position $x$ and $\hslash$ is the reduced Planck's constant.

Solving Quantum Bound States

Specifically, when dealing with quantum bound states like in the textbook exercise, this equation is essential to deducing the number and nature of these states, which are the permissible energy levels of particles confined within a potential well.

Quantum Well

A quantum well is a potential energy configuration in which a particle such as an electron is confined to a region of space by potential barriers that are high compared to the energy of the particle. These wells are often used in semiconductor physics to create areas where charge carriers are confined.

In the context of the exercise, the quantum well is finite, which means that the potential barriers are not infinite. Consequently, this leads to a situation where only a discrete, finite number of energy states are permitted, known as bound states. These bound states, due to their quantized nature, are where a particle can 'reside' with set energy levels as opposed to the continuum of states available in free space.

Potential Energy

Potential energy in quantum mechanics is the energy held by an object because of its position relative to other objects. Within quantum wells, the potential energy describes the 'depth' of the well and represents an integral part of the Schrödinger equation, dictating how particles behave within the well.

In the problem provided, $U_0$ represents the height of the potential well, and this value, coupled with the well's width $L$ and the mass of the particle $m$ being considered, determines the number of bound states allowable in the well. An understanding of potential energy is crucial for grasping important quantum mechanics concepts such as tunneling, where particles can pass through potential barriers, defying classical predictions.