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Lecture 12
Physics 486, Spring ‘
Lecture 12
The Quantum Harmonic
Oscillator: The Operator Method
Harmonic Oscillator: Operator Method
0
2 2 2 2 2
d u x m x u x Eu x m dx
We can rewrite this more concisely in terms of the
operators p and x:
z We’ll now solve the SEQ for the
harmonic oscillator potential using a
more elegant operator method.
Consider the SEQ for the HO potential:
V→∞ V→∞
V(x)
x
p m x u Hu Eu m
(^) + ω = =
where:
d
p
i dx
H p m x m
= + ω and
Harmonic Oscillator: Operator Method
0
However, this won’t work in general for operators – because
operators don’t commute in general - and it won’t work specifically
in this case because we know that p and x don’t commute!
z Now, if the term in brackets […]
consisted of just numbers rather than
operators, we could simply factor it as
follows:
V→∞ V→∞
V(x)
x
However, it interesting to see what we get if we try to
factor (*) in this manner. Let’s define the following factors:
2 2
p + ( m ω x ) = ( ip + m ω x )( − ip + m ω x ) (*)
1/ 2
a ip m x
m
ω
ω
+ =^ −^ +
AND
1/ 2
a ip m x
m
ω
ω
− =^ +
Where a (^) + and a- are understood to be operators.
Harmonic Oscillator: Operator Method
0
z So, we can write:
V→∞ V→∞
V(x)
x
Where we have used the definition of the commutator above:
a a ip m x ip m x
m
ω ω ω
− + =^ +^ −^ +
p m x im xp px
m
ω ω ω
( ) [ , ]
i
p m x x p
m
ω ω
[ , x p ] = i =
[ , x p ] = xp − px
Lecture 12
Harmonic Oscillator: Operator Method
0
z But, we know that the commutator
between x and p is given by:
V→∞ V→∞
V(x)
So, we can write: x
( ) [ , ]
i
a a p m x x p
m
ω ω
− + =^ +^ −
[ , x p ]= i =
As:
a a p m x
m
ω ω
− + =^ +^ +
a a H
ω
− + =^ +
which can be written more succinctly:
Harmonic Oscillator: Operator Method
0
z Consequently, our original Hamiltonian
operator:
V→∞ V→∞
V(x)
can be written as follows: x
H p m x m
= + ω
H ω a a − +
1/ 2
a ip m x
m
ω
ω
AND
1/ 2
a ip m x
m
ω
ω
− =^ +
where, again:
Harmonic Oscillator: Operator Method
0
z Notice that if we reverse the order of
a+ and a - , we get:
V→∞ V→∞
V(x)
x
Consequently, the Hamiltonian operator for
the harmonic oscillator potential can ALSO
be written as follows:
It also follows from these 2 versions of the Hamiltonian, that
a- and a+ have the following commutation relation:
a a p m x
m
ω ω
+ − =^ +^ −
a a H
ω
H ω a a
[ a − , a + ] = 1
Harmonic Oscillator: Operator Method
0
z But what is the significance of the
operators a+ and a -?
It turns out that if u is an eigenstate
of the time-independent SEQ with
eigenvalue E, i.e., Hu=Eu, then it is
ALSO true that:
V→∞ V→∞
V(x)
x
Let’s prove this!
H ( a u + ) = ( E + = ω )( a u +) AND H ( a u − ) = ( E − = ω)( a u −)
( ) ( )
H a u + ω a a + (^) − a u + ω a a a + − (^) + a (^) + u
ω a (^) + a a − (^) + u a (^) + ω a a + (^) − u a (^) + ω a a + (^) − ω u
^
= a + (^) ( H + = ω (^) ) u = a (^) + ( E + =ω (^) ) u = (^) ( E +=ω)( a u +)
H ( a u + ) = ( E + = ω)( a u +)
Lecture 12
Harmonic Oscillator: Operator Method
0
z What is the energy eigenvalue
associated with the lowest
(“ground”) eigenstate,
V→∞ V→∞
V(x)
x
(^2) / 2 e
m x uo x A
− ω
=
We can determine this by applying
the Hamiltonian operator to this
eigenstate:
o
a u
−
E o = = ω
Hu o ω a a + − uo E uo o
Using the fact that:
We find that that:
Harmonic Oscillator: Operator Method
0
z Having established the lowest
(“ground”) eigenstate, V→∞ V→∞
V(x)
x
(^2) / 2 e
m x o
u x A
− ω
=
we can now use the raising operator to
generate the other eigenstates, as follows:
n
un x = An a + uo x
Example – Let’s generate the first excited eigenstate:
( ) ( )
2
1/ 4 1 / 2 1 1 1/ 2 e 2
m x o
A d m u x A a u m x m dx
ω ω ω ω π
−
=
= =
2
1/ 4 1/ 2 / 2 1 1
e
m m m x
u x A x
ω ω ω
π
=
with
E n n ω
Harmonic Oscillator: Operator Method
0
V→∞ V→∞
V(x)
x
2
1/ 2 (^2 2) / 1
2 e 1
m m m x A x dx
ω ω ω
π
∞ −
−∞
=
= =
1/ 2 1/ 2 2
1
2 1 2
m m A m m
ω ω π
π ω ω
=
= =
= =
A 1 (^) = 1
Now we need to find A 1 by normalizing this
eigenstate:
Harmonic Oscillator: Operator Method
Actually, it turns out that we can also determine the
normalization condition using operator methods.
For example, we know that:
( ) ( )
(^2) * * cn un (^) 1 u n (^) 1 dx a un a un dx
∞ ∞
−∞ −∞
Because a (^) + and a- are hermitian conjugates*, we can write
the second integral:
( ) ( ) ( )
a u n a un dx a a un u dxn
∞ ∞
*Note, operators a+ and a (^) - are called hermitian conjugates if for any functions f(x) and g(x): f ( ag ) dx ( af ) gdx
∞
−∞
∞
−∞
± =^
∓
But recall:
ω a a − + un E un n
−^ =
where = ω
En n
a u + n = c un n + 1 where c (^) n is a
normalization constant
Consider: (^) (*)
Lecture 12
Harmonic Oscillator: Operator Method
Rearranging, one finds that:
Consequently, we can rewrite the integral:
( ) ( ) ( ) ( )
- (^) * a un a un dx a a un u dxn n 1 u u dxn n
∞ ∞ ∞
∫ =^ ∫ =^ + ∫
But, because both ψn and ψn+1 are normalized, the above
result implies: 1
2 cn = n +
a a u − + n = ( n + 1 ) un
So, we finally, we can write our original integral (*):
( ) ( ) ( )
(^2) * * * cn un (^) 1 un (^) 1 dx a un a un dx n 1 u u dxn n
∞ ∞ ∞
−∞ −∞ −∞
( )
1/ 2 cn = n + 1
So: (^) ( )
1/ 2 a u + n = n + 1 un + 1 Using a similar calculation: (^) ( )
1/ 2 a u − n = n un − 1
Notice that the u (^) n are NOT eigenstates of a (^) + or a- , and
therefore we don’t expect either a+ or a- to commute with H!
Harmonic Oscillator: Operator Method
Therefore:
u 1 = a u + o
So, we can summarize:
( ) ( )
( )
2 (^2) 1/ 2 1 1/ 2
u = a u + = a + uo
( ) ( )
( )
3 3 1/ 2 2 1/ 2
u = a u + = a (^) + uo ⋅
( ) ( )
( )
4 (^4) 1/ 2 3 1/ 2
u = a u + = a (^) + uo ⋅ ⋅
( )
1/ 2(^ )
n un a uo n
etc., etc…
normalization constant
Harmonic Oscillator: Number Operator
0
z Another important operator to
define is a + a-. To see the effect
of this operator, recall first that:
V→∞ V→∞
V(x)
x
is called the “number” operator.
1/ 2 1/ 2 1/ 2
a a u + − n = a + a u − n = n a u + n − 1 = n n − 1 + 1 un = nun
Consequently:
1/ 2
a u + n = n + 1 un + 1 ( )
1/ 2
a u − n = n un − 1
Nu n = a a u + − n = nun
When operating on eigenstate un , the number operator
generates the quantum number n.
Note that the result (*) above shows that the un are
eigenstates of the number operator as well as the
Hamiltonian H, and hence we expect that N should commute
with H.
(*)
n=3 =ω
7 2
Energy
0
n=
n=1 =ω
n=2 =ω
3 2
5 2
=ω
1 2
= (n+ )=ω
1 2
En
n = 0,1,2,…
What is the meaning and use of the number
operator? As shown by the spectrum to the right,
the quantum mechanical oscillator can be viewed as
having a single oscillatory mode characterized by
the angular frequency ω, but with energy quantized so that En ==ω(n+1/2). If we associate a particle
with each quanta, then n represents the number of
particles in a particular mode. Consequently, the
different eigenstates un represent states of the
system with different numbers of particles. The
number operator “projects out” the number of
particles (“quanta”) in a given state of the system.
Harmonic Oscillator: Number Operator
This description is quite useful in describing, e.g., quantized light
(photons) in a laser cavity - as the light emission is usually dominated by
a single frequency ω - and quantized vibrational modes (“phonons”) in a
solid. In these cases, the operator a (^) + creates a photon/phonon, while a- “annihilates” a photon/phonon. This perspective is the key building
blocks of quantum field theory (see Physics 580, 460 or 560).
Lecture 12
Harmonic Oscillator: Operator Method
and
dV T x dx
V n E n H
ω = (^) + (^) = =
Proof: Let’s calculate the time derivative of the
expectation value :
Note that this implies that:
V = T
for the harmonic oscillator. This is just an example of the quantum
mechanical version of the Virial Theorem, which expresses that (in
one dimension):
[ , ] [ , ] { [ , ] [ , ] }
d xp i i i H xp xp H x p H x H p dt
Recall that we calculated both [H,p] and [H,x] in Lecture 9:
[ ,^ ] [ ,^ ]
i pop x H H x m
and [^ ,^ ]^ [^ ,^ ]
dV p H H p i dx
Harmonic Oscillator: Operator Method
Consequently:
{ [^ ]^ [^ ] }
2
, , 2 2
d xp (^) i dV p x p H x H p x dt dx m
But, for a stationary state, is just a constant (related to
Heisenberg uncertainty principle), so its time-derivative is 0.
Therefore:
[ ]
n
dV T x m x m n n E H dx m
ω ω ω ω
d xp dV x T dt dx
dV T x dx
Note, for the 1D harmonic oscillator (V(x) = ½mω^2 x^2 ):
