Question

Refer to a particle of mass $m$ trapped in a half-infinite well, with potential energy given by $$ U(x)=\left\{$$\begin{array}{ll}\mathrm{\infty }& x\le 0\\ \text{0}& 0<x<L\\ {\text{U}}_0& x\ge L\end{array}$$\right.$$ Write solutions to the Schrödinger equation appropriate in the various regions, impose required continuity conditions, and obtain the energy quantization condition:

Short answer

The energy quantization condition for a particle of mass $m$ in a half-infinite well is given by $k\tan (kL)=\alpha$, where $k=\sqrt{\frac{2mE}{{\hslash }^2}}$ and $\alpha =\sqrt{\frac{2m(U_0-E)}{{\hslash }^2}}$ are used to solve the Schrödinger equation.

Step-by-step solution

Frame the Schrödinger Equation

The time-independent Schrödinger equation is given by $$\frac{-{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}+U(x)\psi =E\psi$$ where $\hslash$ is the reduced Planck’s constant, $m$ is the mass of the particle, $\psi$ is the wave function, $U(x)$ is the potential energy and $E$ is the total energy of the system. The particle's behavior is determined by solving this equation with the potential $U(x)$.

Apply the Schrödinger Equation to Region 1 ( $x\le 0$)

Due to the infinite potential energy for this region, we require the wave function $\psi$ to be finite everywhere. So, ${\psi }_1=0$ for this region.

Apply the Schrödinger Equation to Region 2 ( $0<x<L$)

For this region, potential energy $U=0$. This turns the Schrödinger equation into $$\frac{-{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}=E\psi$$. This is a simple harmonic oscillator equation, with solutions of the form ${\psi }_2=A\sin (kx)+B\cos (kx)$, where $k^2=\frac{2mE}{{\hslash }^2}$. Since ${\psi }_1=0$, we require ${\psi }_2$ to be continuous at $x=0$, which implies $B=0$. Thus, the wave function for region 2 is ${\psi }_2=A\sin (kx)$.

Apply the Schrödinger Equation to Region 3 ( $x\ge L$)

For this region, potential energy $U=U_0$, where $U_0>E$. Thus, the Schrödinger equation becomes $$\frac{-{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}+U_0\psi =E\psi$$ which simplifies to $$\frac{-{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}=(U_0-E)\psi$$. From this type of equation, the wave function for region 3 should be of the form ${\psi }_3=C{e}^{\alpha x}+D{e}^{-\alpha x}$, where ${\alpha }^2=\frac{2m(U_0-E)}{{\hslash }^2}$. Due to that the wave function must remain finite as $x$ approaches infinity, $C=0$, and the function for this region becomes ${\psi }_3=D{e}^{-\alpha x}$.

Meet the Continuity Condition

We need to ensure $\psi$ and $\frac{d\psi }{dx}$ are continuous at $x=L$. This gives us two equations: ${\psi }_2(L)={\psi }_3(L)$ and $\frac{d{\psi }_2}{dx}(L)=\frac{d{\psi }_3}{dx}(L)$. From here, we can find the quantization condition: $k\tan (kL)=\alpha$. This is the energy quantization condition for this problem.

Key concepts

Potential Well

The potential well is a fundamental concept in quantum mechanics, pivotal to understanding how particles behave when they are confined in a restricted region of space. In this exercise, we consider a half-infinite potential well, which means that the potential energy is infinite at one end and has a finite value at the other. This restriction effectively traps the particle in a limited region.

In the half-infinite well:

• The potential is infinite for values of $x\le 0$, thus a particle cannot exist there, i.e., the wave function ${\psi }_1=0$.
• Between $0<x<L$, the potential is zero, allowing the particle to exist freely.
• For $x\ge L$, the potential becomes a finite value $U_0$, still restricted but not infinite, influencing the particle's wave function ${\psi }_3$ beyond this point.

The energy states of the particle hinge upon these potential conditions, creating zones where the particle can and cannot exist.

Energy Quantization

Energy quantization emerges from the conditions imposed on the particle's wave functions within the potential well. Unlike classical systems where energy can vary continuously, quantum mechanics requires that only certain discrete energy levels are permissible for particles in confined spaces.

In this problem, due to boundary and continuity requirements at points $x=0$ and $x=L$, the energy levels are quantized. This means that the particle can only possess specific energies, derived from the quantization condition $k\tan (kL)=\alpha$, where:

• $k$ is related to the particle's energy $E$ in the region $0<x<L$.
• $\alpha$ stems from the potential $U_0$ encountered in $x\ge L$.

The requirement for this continuity and boundary alignment translates to the wave function uncertainties manifesting as discrete energy levels. Thus, the particle's energy is split into distinct packets, important for understanding the particle's stability and state transitions.

Wave Functions

Wave functions are a cornerstone of quantum mechanics, offering a complete description of a quantum state of a particle. Mathematically represented as $\psi (x)$, these functions forecast the probability of finding a particle in a specific location.

For each region of the potential well:

• In the region $x\le 0$, since the potential is infinite, the wave function ${\psi }_1=0$ as the particle cannot exist there.
• In the region $0<x<L$, the wave function ${\psi }_2=A\sin (kx)$ is derived from solving the simplified Schrödinger equation where the potential is zero.
• In the region $x\ge L$, the wave function fits the form ${\psi }_3=D{e}^{-\alpha x}$, indicating an exponential decay as the particle virtually tunnels into the potential barrier.

The conditions at the boundaries ensure that these wave functions remain continuous and differentiable across the different regions of the well. This results, through the principles of superposition and interference, in the emergent quantized energy levels we observe in quantum wells.