Question

Is the superposition wave function for the free particle \(\psi(x)=A_{+} e^{+i \sqrt{\left(2 m E / \hbar^{2}\right)} x}+A_{-} e^{-i \sqrt{\left(2 m E / \hbar^{2}\right)} x}\) an eigenfunction of the momentum operator? Is it an eigenfunction of the total energy operator? Explain your result.

Short answer

The given superposition wave function for a free particle is not an eigenfunction of the momentum operator since applying the momentum operator does not result in a proportional output. However, it is an eigenfunction of the total energy operator, as the wave function is proportional to itself when the total energy operator is applied.

Step-by-step solution

Define Momentum Operator And Wave Function

The momentum operator in one-dimensional space is given by: \[ \hat{p} = -i\hbar \frac{d}{dx} \] The given superposition wave function for the free particle is: \[ \psi(x) = A_{+} e^{+i \sqrt{(2 m E / \hbar^{2})} x} + A_{-} e^{-i \sqrt{(2 m E / \hbar^{2})} x} \]

Apply Momentum Operator To Wave Function

To check if the wave function is an eigenfunction of the momentum operator, we apply the momentum operator to the wave function: \[ (-i\hbar \frac{d}{dx})(A_{+} e^{+i \sqrt{(2 m E / \hbar^{2})} x} + A_{-} e^{-i \sqrt{(2 m E / \hbar^{2})} x}) \] After evaluating the derivative: \[ -i\hbar\left(A_{+} \sqrt{\frac{2mE}{\hbar^2}} e^{+i \sqrt{(2 m E / \hbar^{2})} x} - A_{-} \sqrt{\frac{2mE}{\hbar^2}} e^{-i \sqrt{(2 m E / \hbar^{2})} x}\right) \] Simplifying and reorganizing terms: \[ \hat{p} \psi(x) = +\sqrt{2mE} A_{+} e^{+i \sqrt{(2 m E / \hbar^{2})} x} -\sqrt{2 m E} A_{-} e^{-i \sqrt{(2 m E / \hbar^{2})} x} \]

Define Total Energy (Hamiltonian) Operator

The total energy operator (or Hamiltonian operator) for a free particle is given by: \[ \hat{H} = \frac{\hat{p}^2}{2m} \] Using the momentum operator, we can write this as: \[ \hat{H} = \frac{(-i\hbar \frac{d}{dx})^2}{2m} \]

Apply Total Energy Operator To Wave Function

We apply the total energy operator to the wave function: \[ \frac{(-i\hbar \frac{d}{dx})^2}{2m}(A_{+} e^{+i \sqrt{(2 m E / \hbar^{2})} x} + A_{-} e^{-i \sqrt{(2 m E / \hbar^{2})} x}) \] Taking the second derivative of the wave function and multiplying by the constants: \[ \frac{E}{\hbar^2}(A_{+} e^{+i \sqrt{(2 m E / \hbar^{2})} x}+ A_{-} e^{-i \sqrt{(2 m E /\hbar^{2})} x}) \] Simplifying and reorganizing terms: \[ \hat{H} \psi(x) = E (A_{+} e^{+i \sqrt{(2 m E / \hbar^{2})} x} + A_{-} e^{-i \sqrt{(2 m E / \hbar^{2})} x}) \]

Evaluate Results And Analyze Eigenfunction

Now we can see that applying the momentum operator to the superposition wave function results in: \[ \hat{p} \psi(x) \neq p \psi(x) \] Since the result is not proportional to the original wave function, it is not an eigenfunction of the momentum operator. On the other hand, applying the total energy operator to the superposition wave function results in: \[ \hat{H} \psi(x) = E \psi(x) \] Since the result is proportional to the original wave function, it is an eigenfunction of the total energy operator. In conclusion, the given superposition wave function for the free particle is not an eigenfunction of the momentum operator but is an eigenfunction of the total energy operator.

Key concepts

Momentum Operator

In quantum mechanics, the momentum operator is crucial in describing the momentum of a particle. It is denoted by \( \hat{p} \) and is defined as:\[ \hat{p} = -i\hbar \frac{d}{dx} \]This operator acts on wave functions to provide information about a particle's momentum. The role of the operator is akin to differentiating the wave function and multiplying the result by \(-i\hbar\). This converts a wave with spatial information into one with momentum information.

By applying the momentum operator to a wave function, we can test if that function is an eigenfunction of the operator. An eigenfunction will be scaled by a constant, called an eigenvalue, while retaining its original form. If this property holds, the wave function is directly connected to a specific value of momentum.

• Operator form: dictates how |ψ⟩ changes.
• Spectrum: continuous in free space.
• Eigenfunctions: typically plane waves.

Wave Function

The wave function in quantum mechanics serves as a mathematical description of the quantum state of a system. For a free particle, one particular type of wave function can be a superposition:\[ \psi(x) = A_{+} e^{+i \sqrt{(2 m E / \hbar^{2})} x} + A_{-} e^{-i \sqrt{(2 m E / \hbar^{2})} x} \]This expression combines two exponential terms. Each term represents a wave moving in opposite directions and is associated with a coefficient \(A_+\) or \(A_-\).

Wave functions are integral for determining probabilities. The square of a wave function's magnitude gives the likelihood of finding a particle in a particular position. This probabilistic nature is core to quantum theory, revealing the limits of predictability.

• Describes state: full knowledge of system.
• Complex: contains amplitude and phase.
• Normalization: important for probability.

Energy Operator

The energy operator in quantum mechanics, commonly called the Hamiltonian operator, is fundamental in detailing the total energy of a system. For a free particle, the Hamiltonian is:\[ \hat{H} = \frac{\hat{p}^2}{2m} \]This implies that energy is derived from the momentum operator applied twice, reflecting kinetic energy. In this way, it encompasses how momentum contributes to the system's energy.

Applying the Hamiltonian to a wave function helps establish whether the wave is an eigenfunction of the operator. If it results in a scaled version of the original wave function, then it is an eigenfunction, with the scaling factor being the energy eigenvalue. This means the wave function has a definitive energy, crucial for understanding quantum systems.

• Encodes total energy: all forms included.
• Eigenfunctions: have definite energy values.
• Operator power: drives dynamics.

Eigenfunction

An eigenfunction is a special type of function that remains nearly unchanged by the application of an operator, aside from a scaling factor. When dealing with operators like momentum or energy, eigenfunctions play a crucial role because they help identify physical quantities (like momentum or energy values).

If \(\psi(x)\) is an eigenfunction of an operator \(\hat{O}\), applying \(\hat{O}\) yields:\[ \hat{O} \psi(x) = \lambda \psi(x) \]where \(\lambda\) is the eigenvalue associated with the operator. This property simplifies many quantum systems' analysis, as each eigenfunction corresponds to a measurable quantity.

• Simplicity: reduces complex problems.
• Physical measurable: linked to observable.
• Orthogonality: simplifies calculations.

Superposition

The principle of superposition represents one of quantum mechanics' most intriguing aspects. It states that any wave function can be expressed as a combination of eigenfunctions of an operator. In other words, a complex state can be constructed from simple, well-defined states.

The given wave function demonstrates superposition:\[ \psi(x) = A_{+} e^{+i \sqrt{(2 m E / \hbar^{2})} x} + A_{-} e^{-i \sqrt{(2 m E / \hbar^{2})} x} \]Both terms in the superposition represent waves moving in opposite directions, suggesting multiple potential measurements. This principle allows particles to exhibit particle-like and wave-like behaviors simultaneously.

• Combines states: creates richer physics.
• Interference: crucial for quantum features.
• Measurement: collapses state to outcomes.