Use x=√((ħ)/(2 m ω))(A+A^†)=ℓ(A+A^†) to show for a harmonic oscillator that in the energy repres

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Use x=√((ħ)/(2 m ω))(A+A^†)=ℓ(A+A^†) to show for a harmonic...

Use $$ x=\sqrt{\frac{\hbar}{2 m \omega}}\left(A+A^{\dagger}\right)=\ell\left(A+A^{\dagger}\right) $$ to show for a harmonic oscillator that in the energy representation the operator $x$ is $$ x_{j k}=\ell\left(\begin{array}{cccccccc} 0 & \sqrt{1} & 0 & 0 & \ldots & & & & \\ \sqrt{1} & 0 & \sqrt{2} & 0 & & & & & \\ 0 & \sqrt{2} & 0 & \sqrt{3} & \cdots & & & & \\ & & \sqrt{3} & \cdots & & & & & \\ \ldots & & \ldots & & \cdots & & \cdots & & \\ & & & \cdots & 0 & \sqrt{n-1} & \ldots & & \\ & & & & \sqrt{n-1} & 0 & \sqrt{n} & & \\ & & & & & \sqrt{n} & 0 & \sqrt{n+1} & \ldots \\ & & & & \ldots & & \sqrt{n+1} & 0 & \\ \ldots & & \ldots & & \ldots & & \ldots & & \ldots \end{array}\right) $$ Calculate the same entries for the matrix $p_{j k}$.

Use
$$
x=\sqrt{\frac{\hbar}{2 m \omega}}\left(A+A^{\dagger}\right)=\ell\left(A+A^{\dagger}\right)
$$
to show for a harmonic oscillator that in the energy representation the operator $x$ is
$$
x_{j k}=\ell\left(\begin{array}{cccccccc}
0 & \sqrt{1} & 0 & 0 & \ldots & & & & \\
\sqrt{1} & 0 & \sqrt{2} & 0 & & & & & \\
0 & \sqrt{2} & 0 & \sqrt{3} & \cdots & & & & \\
& & \sqrt{3} & \cdots & & & & & \\
\ldots & & \ldots & & \cdots & & \cdots & & \\
& & & \cdots & 0 & \sqrt{n-1} & \ldots & & \\
& & & & \sqrt{n-1} & 0 & \sqrt{n} & & \\
& & & & & \sqrt{n} & 0 & \sqrt{n+1} & \ldots \\
& & & & \ldots & & \sqrt{n+1} & 0 & \\
\ldots & & \ldots & & \ldots & & \ldots & & \ldots
\end{array}\right)
$$
Calculate the same entries for the matrix $p_{j k}$.

Key Concepts

Ladder Operators

Ladder operators, specifically the annihilation operator A and the creation operator A†, are fundamental in the algebraic treatment of the quantum harmonic oscillator. They are used to raise or lower the energy eigenstates and obey the commutation relation [A, A†] = 1. These operators simplify the construction of the Hamiltonian and allow elegant derivations of matrix elements for observables, with clear rules about how they modify the number of quanta in a state.

Harmonic Oscillator Eigenstates

The quantum harmonic oscillator has a discrete set of energy eigenstates labeled by nonnegative integers. These eigenstates form a complete orthonormal basis in Hilbert space. In this energy representation, operators such as position and momentum are represented as matrices whose entries are determined by how the corresponding operator acts on these eigenstates.

Operator Representation Using Ladder Operators

Both the position and momentum operators can be expressed in terms of the ladder operators A and A†. For instance, the position operator is given as a symmetric combination of A and A†, while the momentum operator involves an antisymmetric combination, typically with an imaginary unit factor. This representation allows one to easily compute the matrix elements by applying the known actions of A and A† on the harmonic oscillator eigenstates.

Momentum Operator in Ladder Operator Basis

The momentum operator in the harmonic oscillator is expressed as p = i?(?m?/2)(A† - A). This form implies that when computing matrix elements in the energy basis, the off?diagonal elements will include factors of ±i, reflecting the operator's Hermitian and anti-symmetric contributions compared to the position operator. The calculation parallels that for the position operator, with the sign and imaginary factors arising due to the difference in the combination of ladder operators used.

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