Question

Write a plane-wave function \(\psi(\vec{r}, t)\) for a nonrelativistic free particle of mass \(m\) moving in three dimensions with momentum p, including correct time dependence as required by the Schrödinger equation. What is the probability density associated with this wave?

Short answer

The probability density for a nonrelativistic free particle with momentum p is constant and is given by \(|A|^2\), where A is the normalization constant that can be determined by integrating the probability density over all space such that it is equal to one.

Step-by-step solution

Write the general solution for the plane-wave function

The general solution for the plane-wave function satisfying the Schrödinger equation is given by: \(\psi(\vec{r},t) = A e^{\frac{i}{\hbar} (\vec{p}\cdot\vec{r} - Et)}\) where A is a normalization constant, \(\vec{r}\) is the position, \(\vec{p}\) is the momentum, E is the energy, and t is the time.

Substitute the given mass and momentum into the solution

We are given the momentum p and mass m. The energy E for a nonrelativistic free particle with momentum p is given by: \(E = \frac{p^2}{2m}\) Substituting the values of E and p into the plane-wave function: \(\psi(\vec{r},t) = A e^{\frac{i}{\hbar} (\vec{p}\cdot\vec{r} - \frac{p^2}{2m}t)}\)

Compute the probability density

The probability density, \(|\psi(\vec{r},t)|^2\), is given by the square of the absolute value of the wave function: \(|\psi(\vec{r},t)|^2 = |\psi(\vec{r},t)| * |\psi^*(\vec{r},t)|\) where \(\psi^*(\vec{r},t)\) is the complex conjugate of the wave function. For the plane-wave function, the complex conjugate is given by: \(\psi^*(\vec{r},t) = A^* e^{-\frac{i}{\hbar} (\vec{p}\cdot\vec{r} - \frac{p^2}{2m}t)}\) Now, calculating the probability density: \(|\psi(\vec{r},t)|^2 = A A^* e^{\frac{i}{\hbar} (\vec{p}\cdot\vec{r} - \frac{p^2}{2m}t)} * e^{-\frac{i}{\hbar} (\vec{p}\cdot\vec{r} - \frac{p^2}{2m}t)} = |A|^2\) The probability density of a free particle with momentum p is constant and given by \(|A|^2\). To obtain the correct normalization, we need to integrate this probability density over all space and set it equal to one, which will give us the value of the normalization constant A.

Key concepts

Probability Density

The probability density is a key concept when discussing quantum mechanics, particularly in terms of interpreting wave functions. For a plane-wave function, the probability density is determined by calculating the absolute square of the wave function, denoted as \(|\psi(\vec{r},t)|^2\).

This involves multiplying the wave function \(\psi(\vec{r},t)\) by its complex conjugate \(\psi^*(\vec{r},t)\). Given the form of the plane-wave function \(|\psi(\vec{r},t)|^2 = |A|^2\), this simplifies to a constant value, which represents the likelihood of finding the particle at any given point in space.

• Probability density is not spatially dependent for plane waves.
• The magnitude of the probability density is dictated by the normalization constant \(A\).

The constancy of \(|A|^2\) reflects the free particle’s equal probability of being found anywhere in space, highlighting a fundamental difference from particles in potential wells or other constraints.

Schrödinger Equation

The Schrödinger Equation is a cornerstone of quantum mechanics, used to describe how the quantum state of a physical system changes over time. It is a partial differential equation that helps determine the wave function of a system.

For a nonrelativistic free particle, the time-dependent Schrödinger Equation is expressed as:\[ i \hbar \frac{\partial}{\partial t} \psi(\vec{r}, t) = \hat{H} \psi(\vec{r},t)\]Here, \(\hat{H}\) is the Hamiltonian operator, which for a free particle is given by:\[ \hat{H} = -\frac{\hbar^2}{2m} abla^2\]This equation provides critical insights into wave function behavior over time.

• Time evolution of the wave function is governed by this equation.
• Solution for free particles appears as plane waves, encompassing both position and momentum.

This connection between kinetic energy and wave functions allows one to predict a particle's future behavior statistically.

Nonrelativistic Free Particle

A nonrelativistic free particle refers to a particle that is not influenced by external forces or potentials and moves at speeds much less than the speed of light. This simplifies our calculations by focusing purely on kinetic energy.

The kinetic energy in nonrelativistic quantum mechanics is given by \(E = \frac{p^2}{2m}\), with \(p\) representing momentum and \(m\) the mass of the particle.

• Nonrelativistic implies speeds are small compared to the speed of light, ensuring relativity isn't needed.
• Free particle motion is linear, following a path dictated by its initial conditions.

These simplifications allow us to use wave functions, such as plane waves, to efficiently describe their motion without the complexity of interactions.

Wave Function Normalization

Normalization of a wave function is crucial in ensuring that it aligns with the probabilistic interpretation of quantum mechanics. Normalization involves adjusting the probability density so that the total probability across all space equals one.

In mathematical terms, this is achieved through the integral:\[\int_{-\infty}^{\infty} |\psi(\vec{r},t)|^2 \, d\vec{r} = 1\]This ensures that a particle is certain to be found somewhere in space. For plane waves, normalization requires careful consideration since their probability density is constant over an infinite space.

• Normalization constant \(A\) is determined such that the integral of the probability density over all space is one.
• This condition is necessary to maintain physical realism and probability conservation.

This process is fundamental in all quantum mechanics, ensuring that theoretical models and real-world expectations align.