Show that in terms of a harmonic oscillator's characteristic...

Show that in terms of a harmonic oscillator's characteristic length $\ell \equiv \sqrt{\hbar / 2 m \omega}$ the ladder operators can be written $$ A=\frac{x}{2 \ell}+\ell \frac{\partial}{\partial x} \quad \text { and } \quad A^{\dagger}=\frac{x}{2 \ell}-\ell \frac{\partial}{\partial x} $$ Hence show that the wavefunction of the second excited state is $\langle x \mid 2\rangle=$ constant $\times\left(x^{2} / \ell^{2}-1\right) \mathrm{e}^{-x^{2} / 4 \ell^{2}}$ and find the normalising constant.

Show that in terms of a harmonic oscillator's characteristic length $\ell \equiv \sqrt{\hbar / 2 m \omega}$ the ladder operators can be written
$$
A=\frac{x}{2 \ell}+\ell \frac{\partial}{\partial x} \quad \text { and } \quad A^{\dagger}=\frac{x}{2 \ell}-\ell \frac{\partial}{\partial x}
$$
Hence show that the wavefunction of the second excited state is $\langle x \mid 2\rangle=$ constant $\times\left(x^{2} / \ell^{2}-1\right) \mathrm{e}^{-x^{2} / 4 \ell^{2}}$ and find the normalising constant.

Key Concepts

Ladder Operators

Ladder operators, also known as creation and annihilation operators, are tools used in the quantum harmonic oscillator to move between energy eigenstates. They encapsulate the algebraic structure of the oscillator by expressing the position and momentum operators in a symmetrical form, which simplifies calculations by allowing one to 'step up' or 'step down' the energy levels.

Characteristic Length

The characteristic length ? = ?(?/(2m?)) is a natural length scale for the harmonic oscillator. It sets the scale for the spatial extent of the oscillator’s wavefunctions and simplifies the expressions for operators and eigenfunctions when non-dimensionalized, making the mathematics more transparent.

Excited State Wavefunction Construction

The excited state wavefunctions of the harmonic oscillator are constructed by applying the ladder operators to the ground state. This procedure generates higher energy eigenfunctions that take the form of a polynomial (specifically, Hermite polynomials) multiplied by a Gaussian envelope, reflecting the quantized nature of the oscillator.

Normalization of Quantum States

Normalization in quantum mechanics ensures that the total probability of finding a particle is one. For the harmonic oscillator, finding the normalization constant involves integrating the squared magnitude of the wavefunction, which typically includes Gaussian terms combined with polynomials, to maintain the probabilistic interpretation of the wavefunction.

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