After choosing units in which everything, including ħ=1, the Hamiltonian of a harmonic oscillato

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After choosing units in which everything, including ħ=1, the...

After choosing units in which everything, including $\hbar=1$, the Hamiltonian of a harmonic oscillator may be written $H=\frac{1}{2}\left(p^{2}+x^{2}\right)$, where $[x, p]=\mathrm{i}$. Show that if $|\psi\rangle$ is a ket that satisfies $H|\psi\rangle=E|\psi\rangle$, then $$ \frac{1}{2}\left(p^{2}+x^{2}\right)(x \mp \mathrm{i} p)|\psi\rangle=(E \pm 1)(x \mp \mathrm{i} p)|\psi\rangle $$ Explain how this algebra enables one to determine the energy eigenvalues of a harmonic oscillator.

After choosing units in which everything, including $\hbar=1$, the Hamiltonian of a harmonic oscillator may be written $H=\frac{1}{2}\left(p^{2}+x^{2}\right)$, where $[x, p]=\mathrm{i}$. Show that if $|\psi\rangle$ is a ket that satisfies $H|\psi\rangle=E|\psi\rangle$, then
$$
\frac{1}{2}\left(p^{2}+x^{2}\right)(x \mp \mathrm{i} p)|\psi\rangle=(E \pm 1)(x \mp \mathrm{i} p)|\psi\rangle
$$
Explain how this algebra enables one to determine the energy eigenvalues of a harmonic oscillator.

Key Concepts

Ladder Operators

Ladder operators provide a method for shifting between energy eigenstates of the harmonic oscillator Hamiltonian. They are defined in terms of the position and momentum operators and allow one to raise or lower the energy of a given state by one quantum. This algebraic method simplifies finding the spectrum by iteratively applying these operators, revealing that the energy levels are equally spaced.

Commutation Relations

Commutation relations, particularly the canonical relation between position and momentum, are foundational in quantum mechanics. For the harmonic oscillator, the commutator [x, p] produces the quantum of action and underlies the non-commutative algebra that defines the structure of the creation and annihilation operators. This relationship is crucial in deriving the energy eigenvalues and understanding how operator algebra dictates the state transitions.

Hamiltonian of the Harmonic Oscillator

The harmonic oscillator Hamiltonian is a central example in quantum mechanics that encapsulates a system with quadratic potential. When written in terms of the position and momentum operators (and often expressed using scaled units such that ?=1), it illustrates how energy is quantized. The algebraic manipulation of the Hamiltonian using ladder operators shows that its eigenstates correspond to quantized energy levels with uniform spacing.

Algebraic Method for Energy Eigenvalues

This algebraic method leverages the properties of ladder operators and the associated commutation relations to determine the energy eigenvalues of the harmonic oscillator without directly solving differential equations. By demonstrating how the application of a ladder operator shifts an eigenstate of the Hamiltonian to another eigenstate with an energy increment or decrement, one can derive that the eigenvalues are quantized and equally spaced, corresponding to the physically observed spectrum of the oscillator.

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