Ladder Operators in Harmonic Oscillator (Step by Step) - Algebraic Method & Solved Problems

Summary

The lecture presents ladder operators as an algebraic alternative to solving the quantum harmonic oscillator directly through the Schrödinger differential equation. Because the oscillator’s energy levels are equally spaced, the operators can move a state up or down one energy level at a time.

1. Set up the oscillator and define the operators

For a particle of mass (m) in the potential (V(x)=\tfrac12 m\omega^2x^2), the Hamiltonian is

[ H=\frac{p^2}{2m}+\frac12m\omega^2x^2. ]

The lecture rescales position and momentum into dimensionless operators:

[ X=\sqrt{\frac{m\omega}{\hbar}}\,x, \qquad P=\frac{p}{\sqrt{m\hbar\omega}}, ]

so that

[ H=\frac{\hbar\omega}{2}(X^2+P^2). ]

It then defines the lowering (annihilation) operator and raising (creation) operator:

[ a=\frac{X+iP}{\sqrt2}, \qquad a^\dagger=\frac{X-iP}{\sqrt2}. ]

These operators are adjoints of one another, but they are not equal and are not Hermitian.

2. Derive the operator relations

Using the canonical commutator ([x,p]=i\hbar), the lecture obtains

[ [a,a^\dagger]=1. ]

The Hamiltonian can be written in either ordering:

[ H=\hbar\omega\left(a^\dagger a+\frac12\right) =\hbar\omega\left(aa^\dagger-\frac12\right). ]

The ordering matters because the ladder operators do not commute.

3. Show how the operators change energy states

For an energy eigenstate (|n\rangle):

• (a|n\rangle) is a state with energy lower by (\hbar\omega).
• (a^\dagger|n\rangle) is a state with energy higher by (\hbar\omega).

The normalized forms are

[ a|n\rangle=\sqrt n\,|n-1\rangle, \qquad a^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle. ]

Thus, the lowering operator annihilates the ground state:

[ a|0\rangle=0. ]

4. Number operator and energy spectrum

The number operator is defined as

[ N=a^\dagger a. ]

It acts on an energy state as

[ N|n\rangle=n|n\rangle, ]

so it returns the state’s quantum number. In terms of this operator,

[ H=\hbar\omega\left(N+\frac12\right), ]

which gives the energy levels

[ E_n=\left(n+\frac12\right)\hbar\omega. ]

The half-quantum term means the ground-state energy is nonzero.

5. Find the ground-state wavefunction

Starting from (a|0\rangle=0), the lecture writes the annihilation operator in position space, where (p=-i\hbar\,d/dx). This gives a first-order differential equation for the ground-state wavefunction. Solving and normalizing it yields

[ \psi_0(x)= \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp\left(-\frac{m\omega x^2}{2\hbar}\right). ]

This demonstrates how the ladder-operator method can produce a wavefunction, not just energy relations.

6. Generate excited states

Repeated application of the raising operator builds higher-energy states from the ground state:

[ |n\rangle=\frac{(a^\dagger)^n}{\sqrt{n!}}\,|0\rangle. ]

The normalization factor follows from the (\sqrt{n+1}) coefficient each time the operator raises the state.

7. Express position and momentum using ladder operators

The definitions can be rearranged to write

[ x=\sqrt{\frac{\hbar}{2m\omega}}\,(a+a^\dagger), ]

[ p=i\sqrt{\frac{m\hbar\omega}{2}}\,(a^\dagger-a). ]

These expressions make expectation-value and uncertainty calculations easier because they can be evaluated using the ladder operators’ actions on states, often without explicit integrals over complicated wavefunctions.

8. Example: Position expectation value in a superposition

The worked problem asks for (\langle x\rangle) in the normalized state

[ |\psi\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2}. ]

Substituting the ladder-operator expression for (x), most terms vanish because different energy eigenstates are orthogonal and (a|0\rangle=0). The result is

[ \langle x\rangle=\sqrt{\frac{\hbar}{2m\omega}}. ]

The subtitle transcript appears to omit the square root in its final written answer; the expression above follows from the stated operator relation and calculation.

Speakers and sources featured

• Divya Jyoti Das — lecturer and presenter.
• Elevate Classes — mentioned in a promotional segment for physics preparation courses; no separate speaker is identified.