Simple Harmonic Oscillator Classical harmonic oscillator

Linear force acting on a particle (Hooke’s law):

F = !kx

F = ma = m d 2 x

dt 2 = !kx

" d 2 x

dt 2 +#

2 x = 0, # = k / m

x(t) = x(0)cos(!t) + p(0)

m! sin(!t)

p(t) = m dx

dt = p(0)cos !t( ) " m!x(0)sin(!t)

From Newton’s law:

Position and momentum solutions oscillate in time:

Simple Harmonic Oscillator Classical harmonic oscillator

Classical Hamiltonian

H = T +V

T = p 2

2m

F = ! "V

"x #V =

1

2 kx

2 = 1

2 m$

2 x 2

V(x)

x

H = p 2

2m + 1

2 m!

2 x 2

Simple Harmonic Oscillator Quantum harmonic oscillator

Quantum Hamiltonian: replace x and p variables with operators

H = T +V = p 2

2m + 1

2 m!

2 x 2

Define a dimensionless operator

a = m!

2! x + i

1

2m!! p

Then

a † =

m! 2!

x + i 1

2m!! p

"

#$ %

&'

†

= m! 2!

x ( i 1

2m!! p

Position, momentum operators obey the canonical commutation relation:

[x, p] = i!

Simple Harmonic Oscillator Quantum harmonic oscillator

Commutation relation:

a,a †!" #$ =

m% 2!

x + i 1

2m%! p

&

'( )

*+ ,

m% 2!

x , i 1

2m%! p

&

'( )

*+ !

" - -

#

$ . .

= m% 2! [x, x]+

i

2! [p, x],

i

2! [x, p]+

1

2m%! [p, p]

= i

2! (,i!) ,

i

2! (i!) = 1

a,a †!" #$ = 1

a † ,a!" #$ = %1

Simple Harmonic Oscillator Quantum harmonic oscillator

Number operator:

Hence we can rewrite the Hamiltonian in terms of the number operator:

N = a † a =

m! 2!

x "i 1

2m!! p

#

$% &

'( m! 2!

x + i 1

2m!! p

#

$% &

'(

= 1

!! p 2

2m + 1

2 m! 2x2

# $%

& '( " 1

2

N = a † a =

m! 2!

x " i 1

2m!! p

#

$% &

'( m! 2!

x + i 1

2m!! p

#

$% &

'(

H = p 2

2m + 1

2 m! 2x2 = !! a†a +

1

2

# $%

& '(

Simple Harmonic Oscillator Quantum harmonic oscillator Number operator:

[N ,H ] = a † a,H!" #$ = !% a

† a, a

† a +

1

2

& '(

) *+

!

" ,

#

$ - = 0

[N ,a] = a † a,a!" #$ = a

† a,a[ ] + a†,a!" #$a = %a

[N ,a † ] = a

† a,a

†!" #$ = a † a,a

†!" #$ + a † ,a

†!" #$a = a †

N † = a

† a( )

†

= a † a †( ) †

= a † a = N

Commutation relations

[H ,a] = !! N +1 2,a[ ] = !! N ,a[ ] = "!!a

[H ,a † ] = !! N +1 2,a†#$ %& = !! N ,a

†#$ %& = !!a †

Simple Harmonic Oscillator Quantum harmonic oscillator

Eigenvalues and eigenfunctions The energy eigenfunctions and eigenvalues can be found by analytically solving the TISE. Here we will use operator algebra:

Energy eigenvalue equation (TISE):

H = p 2

2m + 1

2 m! 2x2 = !! N +

1

2

" #$

% &' = !! a†a +

1

2

" #$

% &'

H ! n = E

n ! n

Ha ! n = aH " !#a( ) !

n = E

n " !#( )a !n

Ha † !

n = a

† H + !#a†( ) !n = En + !#( )a

† ! n

Notice that:

The parentheses around ψ are standard (Dirac) notation for states that is independent of x or p representation. More on this notation later.

Simple Harmonic Oscillator Quantum harmonic oscillator

Eigenvalues and eigenfunctions

The state is an energy eigenfunction with eigenvalue

The state is an energy eigenfunction with eigenvalue

H ! n = E

n ! n

Ha ! n = E

n " !#( )a !n

Ha † !

n = E

n + !#( )a† !n

Hence a and a+ are called the raising and lowering (ladder) operators since they raise or lower the energy by a definite amount.

a ! n

a † !

n

E n ! !"( )

E n + !!( )

Simple Harmonic Oscillator Quantum harmonic oscillator

Eigenvalues and eigenfunctions

Consider the lowest eigenvalue of H (ground state energy):

H ! n = E

n ! n

H ! 0 = E

0 ! 0

The lowering operator a cannot lower the energy of this eigenstate any further. Hence,

a ! 0 = 0

!!a†( )a "0 = !!N "0 = H # !! 2

$ %&

' () " 0 = E

0 # !! 2

$ %&

' () " 0 = 0

E 0 = !!

2

Ground state energy:

Simple Harmonic Oscillator Quantum harmonic oscillator

Eigenvalues and eigenfunctions

H ! n = E

n ! n

H ! 0 = E

0 ! 0

We have seen that the states

are energy eigenstates with energy .

Thus starting with the lowest energy E0, the energy eigenvalues are

E 0 ,E

0 + !! ,E

0 + 2!! ,......

E n = E

0 + n!! = n +

1

2

" #$

% &' !!

E 0 = !!

2

a † !

n

E n + !!( )

Simple Harmonic Oscillator Quantum harmonic oscillator

Eigenvalues and eigenfunctions

A unique feature of the quantum harmonic oscillator is that the energy eigenvalues are equally spaced:

E n = n +

1

2

! "#

$ %& !'

Simple Harmonic Oscillator Quantum harmonic oscillator

! 0 (x) = N

0 e " x2 x

c

2

N 0

2 =

m#

$! , x

c =

2!

m#

Consider the ground state:

! m"

2! # 0 + !

2m"

$# 0

$x = 0

Normalized solution:

The lowering operator a cannot lower the energy of this eigenstate any further. Hence,

a ! 0 = 0

Simple Harmonic Oscillator Quantum harmonic oscillator

! 0 (x) = N

0 e " x2 x

c

2

N 0

2 =

m#

$! , x

c =

2!

m#

Now we can calculate the higher energy (excited) states:

! 1 (x) = N

1 a †! 0 (x) = N

1

x

x c

" x c

2

# #x

$

%& '

() ! 0 (x)

! 2 (x) = N

2 a †( ) 2

! 0 (x) = N

2

x

x c

" x c

2

# #x

$

%& '

()

2

! 0 (x)

! n (x) = N

n a †( )

n

! 0 (x) = N

n

x

x c

" x c

2

# #x

$

%& '

()

n

! 0 (x)

Simple Harmonic Oscillator Quantum harmonic oscillator

! 0 (x) = N

0 e " x2 x

c

2

N 0

2 =

m#

$! , x

c =

2!

m#

Now we can calculate the higher energy (excited) states:

! n (x) =

1

2 n n! H

n y( )!0 (x)

Normalized solutions:

Hn(y): Hermite polynomials

y = 2x

x c

Simple Harmonic Oscillator First few Hermite polynomials:

H 0 (x) = 1, H

1 (x) = x, H

2 (x) = 4x

2 ! 2

There are many properties known about Hermite polynomials. See http://mathworld.wolfram.com/HermitePolynomial.html or your favourite mathematics book of special functions for more.

Simple Harmonic Oscillator Quantum harmonic oscillator

Ground State Expectation values (verify this using the ladder operators, a and a+. See Example 2.5 in the textbook)

!p 0 = p

2

0 " p

2

= m#!

2

!x 0 = x

2

0 " x

2

= !

2m#

!x 0 !p

0 = !

2

The ground state is a minimum uncertainty state. Recall that such a state must be Gaussian.