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

Natural scales Energy scale is � ω . Using harmonic oscillator parameters m , � , ω , can we determine natural length scale � � ℓ = m ω This will help in writing dimensionless operator ˆ ℓ . x ˆ p Similarly dimensionless momentum operator will be √ � m ω a † as follows Define operators ˆ a and ˆ ˆ x p ˆ ˆ = + i √ a � 2 � / m ω 2 � m ω ˆ x p ˆ a † ˆ = − i √ � 2 � / m ω 2 � m ω Note they are not hermitean operators! Lecture 20: Harmonic Oscillator (ladder operato JAN-APR 2016 () EP228: Quantum Mechanics I / 1

Position & Momentum operators Position operator is � x = 1 2 � a † ) ˆ 2(ˆ a + ˆ m ω Similarly momentum operator is √ p = 1 a † ) ˆ 2 i (ˆ a − ˆ 2 � m ω Using commutator [ˆ x , ˆ p ] = i � , we can derive a † ] = I [ˆ a , ˆ a † , harmonic oscillator Hamiltonian is In terms of ˆ a , ˆ a + 1 ˆ a † ˆ H = � ω (ˆ 2) Lecture 20: Harmonic Oscillator (ladder operato JAN-APR 2016 () EP228: Quantum Mechanics I / 1

a † a = ˆ 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 � a † ˆ � ψ | 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 a ] , [ˆ a † ] N , ˆ N , ˆ a | λ � is eigenstate of ˆ Show ˆ N with eigenvalue λ − 1 a | λ � is eigenstate of ˆ and ˆ N with eigenvalue λ + 1 Lecture 20: Harmonic Oscillator (ladder operato JAN-APR 2016 () EP228: Quantum Mechanics I / 1

By the way � λ | λ � = � λ − 1 | λ − 1 � = 1. We get the following implications ˆ a | λ � = c | λ − 1 � a † | λ � = d λ | λ + 1 � ˆ Determine c λ , d λ using the above data. Lecture 20: Harmonic Oscillator (ladder operato JAN-APR 2016 () EP228: Quantum Mechanics I / 1