Given that A|En⟩=α|En-1⟩and En=(n+(1)/(2)) ħω, where the annihilation operator of the harmonic o

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Given that A|En⟩=α|En-1⟩and En=(n+(1)/(2)) ħω, where the...

Given that $A\left|E_{n}\right\rangle=\alpha\left|E_{n-1}\right\rangle$ and $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega$, where the annihilation operator of the harmonic oscillator is $$ A \equiv \frac{m \omega x+\mathrm{i} p}{\sqrt{2 m \hbar \omega}} $$ show that $\alpha=\sqrt{n}$. Hint: consider $\left.|A| E_{n}\right\rangle\left.\right|^{2}$.

Given that $A\left|E_{n}\right\rangle=\alpha\left|E_{n-1}\right\rangle$ and $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega$, where the annihilation operator of the harmonic oscillator is
$$
A \equiv \frac{m \omega x+\mathrm{i} p}{\sqrt{2 m \hbar \omega}}
$$
show that $\alpha=\sqrt{n}$. Hint: consider $\left.|A| E_{n}\right\rangle\left.\right|^{2}$.

Key Concepts

Operator Algebra and Commutation Relations

The ladder operators obey specific commutation relations, notably [A, A†] = 1. These relations are instrumental in deriving the properties of the quantum harmonic oscillator, including the normalization factors when operators act on eigenstates, and they underpin the entire algebraic solution of the oscillator.

Normalization and Inner Products

In quantum mechanics, the normalization of state vectors ensures that the total probability is one. By computing the inner product (or squared norm) of a state after the action of an operator like the annihilation operator, one can determine the necessary normalization constants, which in this case leads to the factor ?n when transitioning from |E?? to |E????.

Quantum Harmonic Oscillator

This is a fundamental model in quantum mechanics that describes a particle bound in a quadratic potential. Its energy eigenstates are quantized, with energies given by E? = (n + 1/2)??, and the model serves as the basis for more complex quantum systems.

Ladder Operators

Ladder operators (annihilation and creation operators) are operators that raise or lower the energy state of a system by one quantum. Their algebra simplifies the study of the quantum harmonic oscillator by providing an efficient way to express transitions between eigenstates and naturally introduces the normalization factors, such as the ?n factor that appears when lowering the state.

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