Compute the expectation value of the x component of the...

Compute the expectation value of the $x$ component of the momentum of a particle of mass $m$ in the $n=3$ level of a one-dimensional infinite square well of width $L$. Reconcile your answer with the fact that the kinetic energy of the particle in this level is $9 \pi^{2} \hbar^{2} / 2 m L^{2}$.

Key Concepts

Kinetic Energy in Quantum Systems

In systems like the infinite square well, the kinetic energy is derived from the squared momentum operator and corresponds to the quantized energy levels of the system. The fact that a state has a specific non-zero kinetic energy does not necessarily imply a non-zero expectation mean momentum, owing to the symmetric properties of the stationary state eigenfunctions.

Stationary State Eigenfunctions

These are solutions to the time-independent Schrödinger equation corresponding to specific quantized energy levels. In the context of the infinite square well, the eigenfunctions are sine functions that exhibit specific symmetry properties. These symmetries often result in certain expectation values, such as that of the momentum, being zero even though the particle has non-zero kinetic energy.

Quantum Expectation Value

This concept refers to the statistical average of a measurable quantity represented by an operator in quantum mechanics. The expectation value is computed by taking the integral of the product of the wavefunction, the operator, and the complex conjugate of the wavefunction over all space, which gives the predicted average outcome of many measurements on identically prepared systems.

Momentum Operator

In quantum mechanics, the momentum operator in one dimension is given by -i? (d/dx). Its expectation value involves taking the integral of the wavefunction, the operator acting on the wavefunction, and the complex conjugate of the wavefunction. Despite non-zero kinetic energies (related to the square of the momentum operator), the expectation value of momentum itself may be zero due to the symmetry properties of the wavefunctions.

Infinite Square Well

This is a model system in quantum mechanics where a particle is confined in a potential that is zero within a finite region and infinite outside. The boundary conditions imposed by the infinite potential lead to a discrete set of energy levels and well-defined stationary states, which are typically sinusoidal functions within the well.

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