Question

A particle is bound by a potential energy of the form \(U(x)=\left\\{\begin{array}{cl}0 & |r|<\frac{1}{2} a \\ \infty & |x|>\frac{1}{2} a\end{array}\right.\) This differs from the infinite well of Section 5.5 in being symmetric about \(x=0\), which implies that the probability densities must also he symmetric. Noting that either sine or cosine would fit this requirement and could be made continuous with the zero wave function outside the well. determiner the allowed energies and corresponding normalized wave functions for this potential well.

Short answer

The allowed energies are \( E_n = \frac{\hbar^2 (2n-1)^2 π^2}{2m a^2}\) for \(n = 1, 2, 3, ...\). The corresponding normalized wave functions are \(ψ_n(x) = \sqrt{\frac{2}{a}}\ cos(k_nx)\), where \(n = 1, 2, 3, ....\).

Step-by-step solution

Specify the potential function forms

The particle's potential energy is given to be \(U(x)=0\) for \( |r|<\frac{1}{2} a \) and \(U(x) = \infty \) for \( |x|>\frac{1}{2}a \). This indicates the particle is free to move within a certain central region where the potential energy \(U(x) = 0\), but restricted from moving outside that region (where the potential energy is infinite).

Construct the wave function

For a symmetric potential, the solution to the Schrodinger equation (which will give the wave functions) can either be a sine or a cosine function. However, it also should be continuous with zero value at \( |x| = \frac{1}{2}a \). The cosine function fulfills these requirements, so we choose the wave function as cos(kx), where k is a constant.

Determine the wave number

The wave number k is found by exploiting the boundary condition that the wave function must be zero at \(x = \pm \frac{1}{2}a\). Thus, \(cos(k \frac{a}{2}) = 0\), which gives \(k_n = \frac{(2n−1)π}{a}\) where \(n = 1, 2, 3, ...\). This is the expression for the 'allowed' wave numbers.

Determine the allowed energy levels

The allowed energy levels can be found by substituting these wave numbers into the kinetic energy of the particle inside the well, which is \((\hbar k)^2/(2m)\), where m is the mass of the particle and \(h\) is Planck's constant. This gives \(E_n = \frac{\hbar^2 (2n-1)^2 π^2}{2m a^2}\), where \(n = 1, 2, 3, ...\). These are the 'allowed' energy levels for the particle.

Normalize the wavefunctions

Normalize the wavefunctions to get the final form. The normalization condition requires that the integral over all space of the magnitude squared of the wavefunction (|cos(k_nx)|^2) equals 1, which leads to the normalized wavefunctions \(ψ_n(x) = \sqrt{\frac{2}{a}}\ cos(k_nx)\), where \(n = 1, 2, 3, ....\)

Key concepts

Potential Energy Well

A potential energy well is a fundamental concept in quantum mechanics that describes a region where a particle is trapped. In this specific exercise, the potential energy well is defined by the potential energy function:

• Inside the interval \(-\frac{1}{2}a < x < \frac{1}{2}a\), the potential energy, \(U(x)\), is zero, indicating that the particle can move freely here.
• Outside this region, \(U(x)\) is infinite, meaning the particle cannot exist in those areas.

This creates an effective boundary for the particle, resembling an infinite square well. The potential well is symmetric about the origin \(x = 0\). This symmetry simplifies our analysis, particularly in determining the suitable wave functions for the particle, as both the wave function and its probability density must reflect this symmetry.

Wave Functions

In quantum mechanics, wave functions describe the quantum state of particles, determining the probability of finding a particle in a particular region. The symmetric potential in this exercise implies that either sine or cosine functions can describe the wave functions. However, cosine functions are utilized here for continuity when the wave function approaches zero at the boundary:

• The ideal choice for our wave function, hence, is \(\cos(kx)\), where \(k\) is a wave number determined from boundary conditions.
• This choice ensures that the wave function naturally goes to zero at \(x = \pm \frac{1}{2}a\), aligning with the boundaries of the potential energy well.

A well-chosen wave function allows us to calculate various quantum properties, especially the particle's energy levels mentioned next.

Allowed Energy Levels

The allowed energy levels in a potential well arise because the particle can only have specific standing wave patterns or quantized states. For this exercise:

• The wave number \(k\) is calculated using the boundary condition \(cos(k\frac{a}{2}) = 0\), leading to \( k_n = \frac{(2n-1)\pi}{a}\), where \(n\) is a positive integer representing the quantum number.
• Each \(k_n\) corresponds to a distinct energy level, calculated from the kinetic energy formula \(E_n = \frac{\hbar^2 (2n-1)^2 \pi^2}{2m a^2}\).

These quantized energy levels mean that a particle can only occupy certain energy states within the well, dictated by the quantum number \(n\), and this highlights the discrete nature of energy in quantum systems.

Normalization of Wave Functions

Wave functions must be normalized to maintain accurate probability interpretations in quantum mechanics. Normalization ensures that the total probability of finding a particle across all space equals one. In this exercise:

• The normalization condition is applied to calculate the final form of the wave functions. It results in \( \psi_n(x) = \sqrt{\frac{2}{a}}\ \cos(k_nx)\).
• Normalization involves integrating the square of the wave function over the well's range and setting this equal to one.

By normalizing the wave function, we ensure that the calculated probability density is accurate and consistent, allowing for meaningful physical predictions and insights.