EP228: Quantum Mechanics I JAN-APR 2016 Lecture 20: Harmonic - - PowerPoint PPT Presentation

EP228: Quantum Mechanics I

JAN-APR 2016 Lecture 20: Harmonic Oscillator (ladder operator method)

JAN-APR 2016 () EP228: Quantum Mechanics I Lecture 20: Harmonic Oscillator (ladder operato / 1

Natural scales

Energy scale is ω. Using harmonic oscillator parameters m, , ω, can we determine natural length scale ℓ =

• mω

This will help in writing dimensionless operator ˆ

x ℓ .

Similarly dimensionless momentum operator will be

ˆ p √ mω

Define operators ˆ a and ˆ a† as follows ˆ a = ˆ x

• 2/mω

+ i ˆ p √ 2mω ˆ a† = ˆ x

• 2/mω

− i ˆ p √ 2mω Note they are not hermitean operators!

JAN-APR 2016 () EP228: Quantum Mechanics I Lecture 20: Harmonic Oscillator (ladder operato / 1

Position & Momentum operators

Position operator is ˆ x = 1 2(ˆ a + ˆ a†)

• 2

mω Similarly momentum operator is ˆ p = 1 2i (ˆ a − ˆ a†) √ 2mω Using commutator [ˆ x, ˆ p] = i, we can derive [ˆ a, ˆ a†] = I In terms of ˆ a, ˆ a†, harmonic oscillator Hamiltonian is ˆ H = ω(ˆ a†ˆ a + 1 2)

JAN-APR 2016 () EP228: Quantum Mechanics I Lecture 20: Harmonic Oscillator (ladder operato / 1

ˆ a†a = ˆ N is usually called number operator- we will see why. Take an arbitrary state |ψ. What can we say about the matrix element ψ|ˆ N|ψ ψ|ˆ a†ˆ a|ψ = χ|χ ≥ 0 Eigenstates of number operator ˆ N ˆ N|λ = λ|λ where λ are real non-negative eigenvalues because ˆ N is hermitean and matrix elements are positive definite. By the way, these are eigenstates of ˆ H also. Check out commutators [ˆ N, ˆ a], [ˆ N, ˆ a†] Show ˆ a|λ is eigenstate of ˆ N with eigenvalue λ − 1 and ˆ a|λ is eigenstate of ˆ N with eigenvalue λ + 1

JAN-APR 2016 () EP228: Quantum Mechanics I Lecture 20: Harmonic Oscillator (ladder operato / 1

By the way λ|λ = λ − 1|λ − 1 = 1. We get the following implications ˆ a|λ = c|λ − 1 ˆ a†|λ = dλ|λ + 1 Determine cλ, dλ using the above data.

JAN-APR 2016 () EP228: Quantum Mechanics I Lecture 20: Harmonic Oscillator (ladder operato / 1