EP228: Quantum Mechanics I JAN-APR 2016 Lecture 24: Oscillator algebra applications (charged particle in magnetic field, Schwinger oscillator model of angular momentum) Lecture 24: Oscillator algebra applications (cha JAN-APR 2016 EP228: Quantum Mechanics I / 7
Charged particle in a magnetic field B along z Π 2 H = Π 2 2 m + p 2 Hamiltonian ˆ 2 m + y 2 m where Π i = ( p i − eA i ). x z ˆ a † a | n � = n | n � N | n � = ˆ H | n � = ( n + 1 / 2) � ω | n � We have derived the operation of ladder operators on | n � as follows: √ n | n − 1 � a | n � = √ a † | n � = n + 1 | n + 1 � Lecture 24: Oscillator algebra applications (cha JAN-APR 2016 EP228: Quantum Mechanics I / 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. E n , k z = ( n + 1 / 2) � ω c + � 2 k 2 z 2 where ω c = eB m How to we obtain the eigenvalues using ladder operators? Evaluate the following commutators p . ˆ r ) − ˆ [Π x , Π y ] , ˆ A ( � A ( � r ) . ˆ p Take a gauge (Coulomb gauge) ∇ . A = 0. In this gauge, [Π x , Π y ] = i � eB This is similar to [ x , p x ] commutator which helped to write ladder operators a , a † Lecture 24: Oscillator algebra applications (cha JAN-APR 2016 EP228: Quantum Mechanics I / 7
We do a similar thing here Π x + i Π y ˆ = √ b 2 eB � Π x − i Π y ˆ b † √ = 2 eB � 1 ˆ b † ˆ 2 eB � { Π 2 x + Π 2 = y + i [Π x , Π y ] } b + p 2 ( b † b + 1 / 2) � eB z = H m 2 m It is similar to solving harmonic oscillator with angular frequency ω c = eB m To determine the position space wavefunctions, we need to choose a form for the vector potential A ( � r ) Lecture 24: Oscillator algebra applications (cha JAN-APR 2016 EP228: Quantum Mechanics I / 7
Heisenberg Equation of motion Please work out d ˆ x i / dt in Heisenberg picture: d ˆ dt = 1 x i H ] = p i − eA i x i , ˆ i � [ˆ m Using this result, work out x 2 md ˆ dt 2 = m 1 i � [ d ˆ x i dt , ˆ i H ] = Lorentz force F i By the way quantum mechanical form for Lorentz force will be E + 1 � d r B × d r � ˆ F = e ˆ dt × � B − � 2 dt Lecture 24: Oscillator algebra applications (cha JAN-APR 2016 EP228: Quantum Mechanics I / 7
Angular momentum algebra We have seen the orbital angular momentum ˆ L i and spin angular momentum ˆ S i obey the same algebra [ L i , L j ] = i � ǫ ijk L k ; [ S i , S j ] = i � ǫ ijk S k ; [ L i , S j ] = 0 ; [ L . L , L i ] = [ S . S , S i ] = 0 We will denote angular momentum as J i from now on [ J i , J j ] = i � ǫ ijk J k ; [ J . J , J i ] = 0 Motivated by ladder operators in harmonic oscillator, we will write non-hermitean operators J ± = J 1 ± iJ 2 Lecture 24: Oscillator algebra applications (cha JAN-APR 2016 EP228: Quantum Mechanics I / 7
Write down the algebra using these ladder operators with J 3 [ J + , J − ] = 2 � J 3 ; [ J + , J 3 ] = − � J + ; [ J − , J 3 ] = � J − The maximal compatible set of operators from angular momentum algebra is J . J , J 3 and we can write simultaneous eigenstates as | jm � J 2 | jm � = j ( j + 1) � 2 | jm � ; J 3 | jm � = m � | jm � We will show that J + is raising operator and J − is the lowering operator J + | jm � ∝ | j m + 1 � ; J − | jm � ∝ | jm − 1 � Need to determine the proportionality constant using the above algebra. Lecture 24: Oscillator algebra applications (cha JAN-APR 2016 EP228: Quantum Mechanics I / 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 ± , J 3 is as follows � a † b J + = � ab † = J − Work out [ J + , J − ] = 2 J 3 to determine J 3 J 3 = � 2( a † a − b † b ) = � 2( N a − N b ) What is the form of J . J in terms of the two harmonic oscillator ladder operators? J 2 = � 2 ( N a + N b ) � N a + N b � + 1 2 2 Lecture 24: Oscillator algebra applications (cha JAN-APR 2016 EP228: Quantum Mechanics I / 7
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