Question

Exercises \(90-92\) refer to a particle described by the wave function $$ \psi(x)=\sqrt{\frac{2}{\pi}} a^{3 / 2} \frac{1}{x^{2}+a^{2}} $$ (a) Taking the particle's total energy to be 0 , find the potential energy. (b) On the same axes, sketch the wave function and the potential energy. (c) To what region would the particle be restricted classically?

Short answer

a) The potential energy \( V(x) \) is 0. b) The wave function \( \psi(x) \) decreases symmetrically from a peak at \( x = 0 \) as \( x \) increases or decreases, while the potential energy is represented by a horizontal line on x-axis. c) The particle can exist anywhere along the x-axis, as there are no classically forbidden regions.

Step-by-step solution

(a) Finding the potential energy

In quantum mechanics, the total energy of a particle is given by the sum of its kinetic energy and potential energy. Here it's given that the total energy is 0, so we know that the potential energy equals the kinetic energy. And as the kinetic energy equals the potential energy, in this case, the potential energy \( V(x) \) also equals 0.

(b) Sketching the wave function and potential energy

The wave function \( \psi(x) \) is given in the problem, which goes to zero as \( x \) approaches infinity and as \( x \) approaches negative infinity. Also, the function has one peak point at \( x = 0 \). Considering these attributes, a plot of the wave function can be sketched. On the same axes, since the potential energy is zero throughout the x-axis, it is represented by a horizontal line intersecting the y-axis at zero.

(c) Identifying classically restricted region

In classical mechanics, a particle is restricted to regions where its total energy is greater than the potential energy. Since in this case the total energy of the particle is zero, the particle would be restricted to the region where the potential energy is less than or equal to zero. Given that the potential energy is zero everywhere, there are no classically forbidden regions – the particle can exist everywhere.

Key concepts

Wave Function

In quantum mechanics, the wave function is a crucial concept that describes the quantum state of a particle or system. It is represented by the Greek letter \( \psi \), commonly written as \( \psi(x) \). This function provides the probability amplitude for a particle's position in space. When you square the wave function, \( |\psi(x)|^2 \), you get the probability density, which tells you how likely you are to find the particle at a certain position.

The wave function in the given exercise is \( \psi(x) = \sqrt{\frac{2}{\pi}} a^{3 / 2} \frac{1}{x^{2}+a^{2}} \). It is continuous and differentiable, indicating that the probability density also behaves smoothly across all positions. Notably, as \( x \) approaches infinity or negative infinity, the wave function tends toward zero, suggesting the probability of finding the particle far from the center is negligible.

• The wave function peaks at \( x = 0 \), where the probability of finding the particle is highest.
• Because it represents probability, the wave function must be normalized, ensuring the total probability of finding the particle anywhere is one.

Potential Energy

Potential energy in quantum mechanics refers to the potential energy landscape through which a particle moves. It is a function of position, indicated as \( V(x) \). The potential energy helps determine the allowed and forbidden regions for a particle.

In this exercise, since the total energy is zero, the sum of the particle's kinetic and potential energy is zero. Given this information, we find that both kinetic and potential energy are zero individually across all positions, \( V(x) = 0 \).

• This neutrality results in a flat potential energy graph, represented by a horizontal line on the x-axis.
• This implies there are no barriers restricting the movement of the particle, allowing it to move freely.

Kinetic Energy

Kinetic energy in quantum mechanics represents the energy a particle has due to its motion. It is derived by subtracting the potential energy from the total energy. In classical terms, kinetic energy is \( \frac{1}{2}mv^2 \), but for a quantum particle, it also stems from the wave characteristics.

Considering the exercise, the total energy provided is zero. Because the potential energy is zero, the kinetic energy also turns out to be zero. This balance is key in understanding the dynamics of the quantum particle.

• Zero kinetic energy indicates that under classical interpretations, the particle appears to be motionless.
• However, due to the nature of quantum mechanics, we cannot assume the particle is at rest or has zero velocity, as its wave function spans across positions.

Classical Mechanics

Classical mechanics deals with the motion of particles under the influence of forces in the macroscopic world. It uses laws and equations established by physicists like Newton.

In the context of this exercise, classical mechanics helps identify the regions where a particle would move based on energy constraints. Typically, a particle is confined to regions where its total energy exceeds the potential energy.

• Here, with both total and potential energies being zero, classical constraints don't apply.
• There are no forbidden regions, meaning the particle can exist anywhere along the x-axis.
• This scenario highlights a stark difference between classical and quantum mechanics, where in classical terms, there'd typically be limitations based on potential energy barriers.