Question

Are the eigenfunctions of \(\hat{H}\) for the particle in the one-dimensional box also eigenfunctions of the position operator \(\hat{x} ?\) Calculate the average value of \(x\) for the case where \(n=3 .\) Explain your result by comparing it with what you would expect for a classical particle. Repeat your calculation for \(n=5\) and, from these two results, suggest an expression valid for all values of \(n\). How does your result compare with the prediction based on classical physics?

Short answer

The eigenfunctions of the Hamiltonian operator, \(\hat{H}\), for a particle in a one-dimensional box are not eigenfunctions of the position operator, \(\hat{x}\). For the cases when \(n=3\) and \(n=5\), the average value of \(x\) is given by \(\langle x \rangle = \frac{L}{2}\), which agrees with the classical physics prediction. In general, the expectation value of \(x\) will be the same for all odd values of \(n\) and undefined for even values of \(n\), due to symmetry.

Step-by-step solution

Identify the eigenfunctions of the Hamiltonian operator

The eigenfunctions of the Hamiltonian operator, \(\hat{H}\), for a particle in a one-dimensional box are given by the following wave function: \[ \psi_n(x)=\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L} \right) \] Where \(L\) is the length of the box, \(n\) is an integer (quantum number), and \(\psi_n(x)\) describes the probability density of finding the particle in a specific location within the box.

Consider the position operator eigenfunctions

For a wave function to be an eigenfunction of the position operator, \(\hat{x}\), it must satisfy the following eigenvalue equation: \[\hat{x}\psi(x) = x\psi(x)\] Now, let's examine the wave function for \(n=3\) and see if it is an eigenfunction for the position operator.

Check if the wave function is an eigenfunction of the position operator

Let's use the wave function for \(n=3\): \[\Psi_{n=3}(x)=\sqrt{\frac{2}{L}}\sin \left(\frac{3\pi x}{L} \right)\] Now, let's apply the position operator \(\hat{x}\) on \(\Psi_{n=3}(x)\): \[\hat{x}\Psi_{n=3}(x) = x\sqrt{\frac{2}{L}}\sin \left(\frac{3\pi x}{L} \right)\] The resulting function is clearly different from the original wave function, and there is no scalar eigenvalue satisfying the eigenvalue equation. Therefore, the wave function \(\Psi_{n=3}(x)\) is not an eigenfunction of the position operator.

Compute the average value of \(x\) for \(n=3\)

The average (expectation) value of \(x\) for a given wave function can be calculated using the following formula: \[\langle x \rangle = \int_{0}^{L}x |\Psi_n(x)|^2dx\] For the case when \(n=3\): \[\langle x \rangle = \int_{0}^{L}x \left| \sqrt{\frac{2}{L}}\sin \left(\frac{3\pi x}{L} \right) \right|^2 dx\] Simplifying and carrying out the integration, we find: \[\langle x \rangle = \frac{L}{2}\] This result implies that, on average, the particle will be found at the center of the box.

Compute the average value of \(x\) for \(n=5\)

For the case when \(n=5\), we similarly calculate the expectation value of \(x\): \[\langle x \rangle = \int_{0}^{L}x \left| \sqrt{\frac{2}{L}}\sin \left(\frac{5\pi x}{L} \right) \right|^2 dx\] And after performing the integration, we find: \[\langle x \rangle = \frac{L}{2}\]

Propose a general expression for the average value of \(x\)

From our calculations for both \(n=3\) and \(n=5\), we have found that the expectation value of \(x\) is given by: \[\langle x \rangle = \frac{L}{2}\] We can now suggest that the expectation value of \(x\) will be the same for all odd values of \(n\). In general, we can write: \[ \langle x \rangle = \begin{cases} \frac{L}{2} &\mbox{ for odd } n \\ \text{Undefined} & \mbox{ for even } n, \text{ due to symmetry} \end{cases} \]

Compare the result with the classical prediction

According to classical physics, a classical particle would oscillate back and forth inside the box and spend equal amounts of time at different locations. The average position of the classical particle would also be at the center of the box, which is given by \(\frac{L}{2}\). The results obtained from our quantum mechanical calculations for odd values of \(n\) are consistent with the classical prediction, showing that, on average, the particle is located at the center of the box.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes nature at the smallest scales, such as atoms and subatomic particles. Unlike classical mechanics, which can predict the exact position and momentum of a particle, quantum mechanics only offers the probability of finding a particle in a certain state. This is represented through the wave function, typically denoted as \( \psi \), which contains all the information about a quantum system. The square of the absolute value of the wave function, \( |\psi|^2 \), provides the probability density of finding a particle at a particular location within certain constraints, like a one-dimensional box in our exercise.

In essence, quantum mechanics introduces the concept of quantization, where certain properties, such as energy, can only take on discrete values. This leads to numerous phenomena that are counterintuitive from a classical perspective, such as particles being in multiple states at once (superposition) or affecting each other instantaneously over long distances (entanglement).

Eigenfunctions of Hamiltonian

In the context of quantum mechanics, the Hamiltonian operator \( \hat{H} \) is a crucial concept that corresponds to the total energy of the quantum system. Its eigenfunctions, also known as stationary states, have energies that are specific and unchanging over time. These eigenfunctions are the solutions to the Schrödinger equation, which is a fundamental equation describing how the quantum state of a physical system changes over time.

For a particle in a one-dimensional box, the eigenfunctions \( \psi_n(x) \) are sinusoidal functions that reflect the particle’s allowed states within the box. These states have quantized energies dependent on a quantum number \( n \). It's critical for students to appreciate that the concepts of eigenfunctions and eigenvalues relate to measurable quantities in quantum mechanics. Furthermore, these eigenfunctions are orthogonal to one another, meaning they obey specific mathematical properties that lead to the independence of these states.

Position Operator

The position operator \( \hat{x} \) in quantum mechanics is associated with the measurement of the position of particles. When an operator acts on the wave function and returns that wave function multiplied by a constant (known as an eigenvalue), the wave function is an eigenfunction of the operator.

In the problem at hand, the wave functions for a particle in a one-dimensional box are not eigenfunctions of the position operator. Applying the position operator to these wave functions essentially multiplies the functions by \( x \) without returning a constant multiple, indicating that they do not fit the criteria for eigenfunctions. This demonstrates a key difference between the measured quantities in classical and quantum mechanics, where quantum measurements are tied to the probability distributions described by the wave functions.

Expectation Value

The expectation value in quantum mechanics is the average value of a physical quantity that we would expect to measure over many trials of a quantum experiment. It is computed by taking a probability-weighted average of the physical quantity across all possible outcomes. The expectation value is often denoted as \( \langle Q \rangle \) for a quantity \( Q \).

In the exercise, the expectation value of the position \( \langle x \rangle \) represents the average position where we could expect to find the particle if we were to measure its position repeatedly while in a given state. For odd values of \( n \) in our particle-in-a-box scenario, the expectation value of \( x \) is \( \frac{L}{2} \) which is analogous to a classical particle's average position within the box over time. This result is aligned with the symmetry of the probability distribution for the particle's position and fortuitously reflects classical intuition despite the foundational differences between classical and quantum mechanics.