Use purely matrix methods to show that if \(x_{n m} \neq 0\) for a harmonic oscillator, then \(E_{n}-E_{m}=\pm \hbar(K / \mu)^{\frac{1}{2}}\). Note that for a harmonic oscillator, \(H=p^{2} / 2 \mu+\frac{1}{2} K x^{2}\) \(x p-p x=i \hbar\)

The Hamiltonian is a fundamental concept in quantum mechanics, representing the total energy of the system. For a harmonic oscillator, the Hamiltonian is given by: \[ H = \frac{p^2}{2 \mu} + \frac{1}{2} K x^2 \] Here,

• \( p \) is the momentum operator.
• \( \mu \) is the mass of the particle.
• \( K \) is the spring constant.
• \( x \) is the position operator.

The Hamiltonian provides a way to describe the evolution of the system over time. It consists of two main parts: the kinetic energy \( (\frac{p^2}{2 \mu}) \) and the potential energy \( (\frac{1}{2} K x^2) \). Understanding the Hamiltonian is crucial for solving problems related to quantum harmonic oscillators.

In quantum mechanics, the commutation relation between operators is essential for understanding their behavior. For the harmonic oscillator, the position \( x \) and momentum \( p \) operators follow the commutation relation:

\[ x p - p x = i \hbar \] This equation implies that the order in which you multiply these operators matters. To work with matrix elements, we start with the given relation and use matrix elements: \[ x_{nm} p_{mn} - p_{nm} x_{mn} = i \hbar \delta_{nm} \] In this context: - \( x_{nm} \) represents the matrix element of position between states \( n \) and \( m \). - \( p_{nm} \) represents the matrix element of momentum between states \( n \) and \( m \). - \( \delta_{nm} \) is the Kronecker delta, which is 1 if \( n = m \) and 0 otherwise. Commutators help us determine how operators relate and how they affect quantum states.

Eigenvalues of the Hamiltonian, known as energy eigenvalues, represent the possible energy levels of the system. For a harmonic oscillator, the energy eigenvalues are given by:

\[ E_n = (n + \frac{1}{2}) \hbar \omega \] Here, \( n \) is a non-negative integer (0, 1, 2, ...) representing the quantum number, and \( \omega \) is the angular frequency of the harmonic oscillator. For this problem, we set \( \omega = \sqrt{K / \mu} \). To find the difference between energy levels:

\[ E_n - E_m = \hbar \omega \left(n - m\right) \] Given this energy difference, we can express it as:

\[ E_n - E_m = \pm \hbar \sqrt{K / \mu} \] when \( |n - m| = 1 \). This condition arises from the selection rules of a harmonic oscillator, which dictate that transitions typically occur between adjacent levels. This quantum mechanical behavior helps explain phenomena like the quantization of energy in a harmonic oscillator.