EP228: Quantum Mechanics I JAN-APR 2016 Lecture 24: Oscillator - - PowerPoint PPT Presentation

EP228: Quantum Mechanics I

JAN-APR 2016 Lecture 24: Oscillator algebra applications (charged particle in magnetic field, Schwinger oscillator model of angular momentum)

JAN-APR 2016 EP228: Quantum Mechanics I Lecture 24: Oscillator algebra applications (cha / 7

Charged particle in a magnetic field B along z

Hamiltonian ˆ H = Π2

x

2m + Π2

y

2m + p2

z

2m where Πi = (pi − eAi).

ˆ N|n = a†a|n = n|n ˆ H|n = (n + 1/2)ω|n We have derived the operation of ladder operators on |n as follows: a|n = √n|n − 1 a†|n = √ n + 1|n + 1

JAN-APR 2016 EP228: Quantum Mechanics I Lecture 24: Oscillator algebra applications (cha / 7

Energy eigenvalues?

Recall we took trial wavefunction and a choice for vector potential giving us a shifted harmonic oscillator potential in the y direction leading to Landau levels. En,kz = (n + 1/2)ωc + 2k2

z

2 where ωc = eB

m

How to we obtain the eigenvalues using ladder operators? Evaluate the following commutators [Πx, Πy], ˆ p.ˆ A( r) − ˆ A( r).ˆ p Take a gauge (Coulomb gauge) ∇.A = 0. In this gauge, [Πx, Πy] = ieB This is similar to [x, px] commutator which helped to write ladder

• perators a, a†

JAN-APR 2016 EP228: Quantum Mechanics I Lecture 24: Oscillator algebra applications (cha / 7

We do a similar thing here ˆ b = Πx + iΠy √ 2eB ˆ b† = Πx − iΠy √ 2eB ˆ b†ˆ b = 1 2eB{Π2

x + Π2 y + i[Πx, Πy]}

H = (b†b + 1/2)eB m + p2

z

2m It is similar to solving harmonic oscillator with angular frequency ωc = eB m To determine the position space wavefunctions, we need to choose a form for the vector potential A( r)

JAN-APR 2016 EP228: Quantum Mechanics I Lecture 24: Oscillator algebra applications (cha / 7

Heisenberg Equation of motion

Please work out dˆ xi/dt in Heisenberg picture: dˆ xi dt = 1 i[ˆ xi, ˆ H] = pi − eAi m Using this result, work out mdˆ x2

i

dt2 = m 1 i[dˆ xi dt , ˆ H] = Lorentz force Fi By the way quantum mechanical form for Lorentz force will be ˆ F = e ˆ E + 1 2 dr dt × B − B × dr dt

• JAN-APR 2016

EP228: Quantum Mechanics I Lecture 24: Oscillator algebra applications (cha / 7

Angular momentum algebra

We have seen the orbital angular momentum ˆ Li and spin angular momentum ˆ Si obey the same algebra [Li, Lj] = iǫijkLk ; [Si, Sj] = iǫijkSk ; [Li, Sj] = 0 ; [L.L, Li] = [S.S, Si] = 0 We will denote angular momentum as Ji from now on [Ji, Jj] = iǫijkJk ; [J.J, Ji] = 0 Motivated by ladder operators in harmonic oscillator, we will write non-hermitean operators J± = J1 ± iJ2

JAN-APR 2016 EP228: Quantum Mechanics I Lecture 24: Oscillator algebra applications (cha / 7

Write down the algebra using these ladder operators with J3 [J+, J−] = 2J3 ; [J+, J3] = − J+ ; [J−, J3] = J− The maximal compatible set of operators from angular momentum algebra is J.J, J3 and we can write simultaneous eigenstates as |jm J2|jm = j(j + 1)2|jm ; J3|jm = m|jm We will show that J+ is raising operator and J− is the lowering

• perator

J+|jm ∝ |j m + 1 ; J−|jm ∝ |jm − 1 Need to determine the proportionality constant using the above algebra.

JAN-APR 2016 EP228: Quantum Mechanics I Lecture 24: Oscillator algebra applications (cha / 7

Schwinger oscillator method

Construction of angular momentum algebra using ladder operators of two independent oscillators:a, a†, b, b† [a, a†] = [b, b†] = I ; rest of the commutators are zero like [a, b] = [a, b†] = 0 Representation for J±, J3 is as follows J+ = a†b J− = ab† Work out [J+, J−] = 2J3 to determine J3 J3 = 2(a†a − b†b) = 2(Na − Nb) What is the form of J.J in terms of the two harmonic oscillator ladder

• perators?

J2 = 2 (Na + Nb) 2 Na + Nb 2 + 1

• JAN-APR 2016

EP228: Quantum Mechanics I Lecture 24: Oscillator algebra applications (cha / 7