By expressing the annihilation operator A of the harmonic...

By expressing the annihilation operator $A$ of the harmonic oscillator in the momentum representation, obtain $\langle p \mid 0\rangle$. Check that your expression agrees with that obtained from the Fourier transform of $$ \langle x \mid 0\rangle=\frac{1}{\left(2 \pi \ell^{2}\right)^{1 / 4}} \mathrm{e}^{-x^{2} / 4 \ell^{2}}, \quad \text { where } \quad \ell \equiv \sqrt{\frac{\hbar}{2 m \omega}} $$

By expressing the annihilation operator $A$ of the harmonic oscillator in the momentum representation, obtain $\langle p \mid 0\rangle$. Check that your expression agrees with that obtained from the Fourier transform of
$$
\langle x \mid 0\rangle=\frac{1}{\left(2 \pi \ell^{2}\right)^{1 / 4}} \mathrm{e}^{-x^{2} / 4 \ell^{2}}, \quad \text { where } \quad \ell \equiv \sqrt{\frac{\hbar}{2 m \omega}}
$$

Key Concepts

Fourier Transform

A mathematical tool used to convert functions between different domains, such as between position space and momentum space. In quantum mechanics, the Fourier transform bridges the gap between the position representation of a wavefunction and its momentum representation, ensuring that the same physical state can be described in either domain.

Quantum Harmonic Oscillator

A fundamental model in quantum mechanics where a particle is subject to a quadratic potential. Its analytically soluble structure leads to well-defined, equally spaced energy eigenstates, making it a cornerstone for understanding more complex quantum systems and for developing approximation methods in quantum physics.

Annihilation Operator

An operator that lowers the energy level of a quantum harmonic oscillator by one quantum. It serves as an essential tool in the algebraic approach to quantum mechanics, as its action on the ground state yields zero, and its successive application generates the higher excited states, thereby simplifying the analysis of the oscillator's spectrum.

Momentum Representation

A formulation of quantum mechanics where states and operators are represented in terms of momentum eigenstates rather than position. This perspective is particularly valuable when problems are naturally expressed in momentum space or when analyzing aspects such as scattering, and it relates directly to the experimental measurement of momentum.

Gaussian Wavefunction

A wavefunction with a Gaussian profile, notably representing the ground state of the quantum harmonic oscillator. It is significant because it minimizes the Heisenberg uncertainty product and maintains its Gaussian form under Fourier transformation, reflecting the symmetric treatment of position and momentum in the underlying quantum system.

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