Simple Harmonic OscillatorSimple Harmonic Oscillator Quantum harmonic oscillator Eigenvalues and eigenfunctions The energy eigenfunctions and eigenvalues can be found by analytically

Simple Harmonic OscillatorClassical harmonic oscillator

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

F = !kx

F = ma = md2x

dt2= !kx

"d2x

dt2+#

2x = 0, # = k / m

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

m!sin(!t)

p(t) = mdx

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

From Newton’s law:

Position and momentum solutions oscillate in time:

Simple Harmonic OscillatorClassical harmonic oscillator

Classical Hamiltonian

H = T +V

T =p2

2m

F = !"V

"x#V =

1

2kx

2=1

2m$

2x2

V(x)

x

H =p2

2m+1

2m!

2x2

Simple Harmonic OscillatorQuantum harmonic oscillator

Quantum Hamiltonian: replace x and p variables with operators

H = T +V =p2

2m+1

2m!

2x2

Define a dimensionless operator

a =m!

2!x + i

1

2m!!p

Then

a†=

m!2!

x + i1

2m!!p

"

#$%

&'

†

=m!2!

x ( i1

2m!!p

Position, momentum operators obey the canonical commutation relation:

[x, p] = i!

Simple Harmonic OscillatorQuantum harmonic oscillator

Commutation relation:

a,a†!" #$ =

m%2!

x + i1

2m%!p

&

'()

*+,

m%2!

x , i1

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 OscillatorQuantum harmonic oscillator

Number operator:

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

N = a†a =

m!2!

x "i1

2m!!p

#

$%&

'(m!2!

x + i1

2m!!p

#

$%&

'(

=1

!!p2

2m+1

2m! 2x2

#$%

&'("1

2

N = a†a =

m!2!

x " i1

2m!!p

#

$%&

'(m!2!

x + i1

2m!!p

#

$%&

'(

H =p2

2m+1

2m! 2x2 = !! a†a +

1

2

#$%

&'(

Simple Harmonic OscillatorQuantum harmonic oscillatorNumber 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 OscillatorQuantum harmonic oscillator

Eigenvalues and eigenfunctionsThe energy eigenfunctions and eigenvalues can be found by analyticallysolving the TISE. Here we will use operator algebra:

Energy eigenvalue equation (TISE):

H =p2

2m+1

2m! 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 isindependent of x or p representation. More on this notation later.

Simple Harmonic OscillatorQuantum 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) operatorssince they raise or lower the energy by a definite amount.

a !n

a† !

n

En! !"( )

En+ !!( )

Simple Harmonic OscillatorQuantum 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 anyfurther. Hence,

a !0= 0

!!a†( )a "0 = !!N "0 = H #!!2

$%&

'()"0= E

0#!!2

$%&

'()"0= 0

E0=!!

2

Ground state energy:

Simple Harmonic OscillatorQuantum 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

E0,E

0+ !! ,E

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

En= E

0+ n!! = n +

1

2

"#$

%&'!!

E0=!!

2

a† !

n

En+ !!( )

Simple Harmonic OscillatorQuantum harmonic oscillator

Eigenvalues and eigenfunctions

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

En= n +

1

2

!"#

$%&!'

Simple Harmonic OscillatorQuantum harmonic oscillator

!0(x) = N

0e" x2 x

c

2

N0

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 anyfurther. Hence,

a !0= 0

Simple Harmonic OscillatorQuantum harmonic oscillator

!0(x) = N

0e" x2 x

c

2

N0

2=

m#

$!, x

c=

2!

m#

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

!1(x) = N

1a†!0(x) = N

1

x

xc

"xc

2

##x

$

%&'

()!0(x)

!2(x) = N

2a†( )2

!0(x) = N

2

x

xc

"xc

2

##x

$

%&'

()

2

!0(x)

!n(x) = N

na†( )

n

!0(x) = N

n

x

xc

"xc

2

##x

$

%&'

()

n

!0(x)

Simple Harmonic OscillatorQuantum harmonic oscillator

!0(x) = N

0e" x2 x

c

2

N0

2=

m#

$!, x

c=

2!

m#

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

!n(x) =

1

2nn!H

ny( )!0 (x)

Normalized solutions:

Hn(y): Hermite polynomials

y =2x

xc

Simple Harmonic OscillatorFirst few Hermite polynomials:

H0(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 yourfavourite mathematics book of special functions for more.

Simple Harmonic OscillatorQuantum harmonic oscillator

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

!p0= p

2

0" p

2

=m#!

2

!x0= x

2

0" x

2

=!

2m#

!x0!p

0=!

2

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