Question

Exercise 53 outlines a procedure for predicting how the quantum-mechanically allowed energies for a harmotic oscillator should depend on a quantum number. In essence, allowed kinetic energies are the particle-in-abox energtes, except the length \(L\) is replaced by the distance between classical tuming points, expressed in terms of E. Apply this procedure to a potenu'al energy of the fonn \(U(x)=-b / x\), where \(b\) is a constant. Assume that at the origin there is an infinitely high wall, making it one turning point, and determine the other tuming pomt in terms of \(E\). For the average potential energy. use its value at half way between the tuming points. again in terms of \(E\). Find an expression for the allowed energies in terms of \(m, b\), and \(n\). (Although threedimensional. the hydrogen atom potential energy is of this form. and the allowed energy levels depend on a quantum number exactly as this simple model predicts.)

Short answer

The allowed energies for the system can be expressed as \(E = -\frac{b^2 \pi^2 \hbar^2 m}{2 n^2}\).

Step-by-step solution

Understanding the provided potential energy function.

Observe the provided potential energy function, \(U(x)=-b / x\), where \(b\) is a constant. This function defines an inversely proportional relationship between potential energy and distance, \(x\). A key piece of the problem is understanding that there is an infinitely high wall at the origin, indicating one turning point.

Find the other turning point.

The energy \(E\) of a quantum system at a turning point is equal to the potential energy \(U\) at that point. So, to find the other turning point, it can be expressed in terms of the energy \(E\) using the given potential energy function. Therefore, solving \(E=U(x)\) gives \(x= -b/E\). This yields another turning point on the other side of the origin.

Find the average potential energy.

The average potential energy is given as its value halfway between the turning points, again expressed in terms of \(E\). Taking the average of the potential energies at these two turning points, the average value is \(U_{avg} = \frac{U(0) + U(-b/E)}{2}\) which simplifies down to \(U_{avg} = \frac{-b}{2x}\).

Find an expression for the allowed energies.

The allowed energies for a quantum system like this can be modeled as \(E = 2 \times U_{avg} + \frac{n^2 \pi^2 \hbar^2}{2 m L^2}\). Substituting \(U_{avg}\) and \(L\) into this equation, makes it possible to give \(E\) entirely in terms of \(m, b,\) and \(n\). This results in the final expression \(E = -\frac{b^2 \pi^2 \hbar^2 m}{2 n^2}\). The allowed energies are categorized by the quantum number \(n\) according to this expression.

Key concepts

Potential Energy

In quantum mechanics, potential energy is essential in understanding the behavior of particles. For the quantum harmonic oscillator, the potential energy function is given by \(U(x) = -\frac{b}{x}\). This formula represents an inverse relationship between the potential energy \(U\) and the distance \(x\). A decrease in distance leads to a more negative potential energy. This characteristic is crucial when the potential energy function features an infinitely high wall at the origin. Here, the wall represents a strong boundary condition affecting the particle's motion.

In this context, the potential energy helps determine such important aspects of a quantum system as the turning points. Potential energy plays a dual role as it indicates both the "location" of quantum turning points and influences the energy states of a system.

Turning Points

Turning points in quantum mechanics are the positions where the total energy of the system is equal to the potential energy. These points indicate where a particle changes direction. In particular, for the function \(U(x) = -\frac{b}{x}\), one turning point is located at the origin, where an infinitely high wall acts as a physical constraint.

By setting the total energy \(E\) equal to the potential energy function \(U(x)\), we can find the second turning point in terms of \(E\): \(x = -\frac{b}{E}\). This calculation shows how deeply interconnected the energy and the geometry of the system are. These turning points, existing at precise values, are integral to understanding the motion and behavior of quantum particles.

Quantum Number

Quantum numbers are integral to describing quantized properties of quantum systems. They denote specific levels of energy states. In our exercise, the quantum number \(n\) influences the allowed energies of the system. Quantization emerges naturally from the boundary conditions imposed on the potential energy function's solutions.

Quantum numbers tell us about various properties such as spin, angular momentum, and energy levels. In the case of a quantum harmonic oscillator, quantization means that not every energy level is permissible; rather, only certain discrete levels (denoted by \(n\)) can be occupied by particles.

Allowed Energies

In quantum mechanics, only specific energy levels are allowed for a particle. These are called 'allowed energies.' For a potential energy function like \(U(x) = -\frac{b}{x}\), the allowed energies are determined by both the potential and the quantized states expressed through quantum numbers.

The expression \(E = -\frac{b^2 \pi^2 \hbar^2 m}{2 n^2}\) illustrates how the allowed energies depend on system parameters such as mass \(m\), constant \(b\), and the quantum number \(n\). These energies are not just arbitrary but emerge from solving the Schrödinger equation under the given boundary conditions dictated by the potential energy function. The concept of allowed energies reflects the nature of quantum systems, where energy levels are discrete and not continuous.

Hydrogen Atom

The hydrogen atom is often used as a model in quantum mechanics to explain fundamental concepts. Its potential energy function bears similarities to those discussed in the exercise. For hydrogen, the potential energy is \(U(x) = -\frac{b}{x}\), paralleling the model used for quantum harmonic oscillators.

The Schrödinger equation for a hydrogen atom, when solved, illustrates how discrete energy levels arise from the quantization of the system. These energy levels mirror those predicted by the quantum number, just like in the harmonic oscillator problem. This illustrative parallel deepens understanding by showing how simple quantum models can mimic more complex atomic behaviors, linking the theory of quantum mechanics seamlessly with real-world atomic structures.