simple harmonic oscillatorsimple harmonic oscillator quantum harmonic oscillator eigenvalues and...
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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:
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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
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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
"
#$%
&'
†
=m!2!
x ( i1
2m!!p
Position, momentum operators obey the canonical commutation relation:
[x, p] = i!
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Simple Harmonic OscillatorQuantum harmonic oscillator
Commutation relation:
a,a†!" #$ =
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
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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
#
$%&
'(
H =p2
2m+1
2m! 2x2 = !! a†a +
1
2
#$%
&'(
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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†
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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.
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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+ !!( )
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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
$%&
'()"0= 0
E0=!!
2
Ground state energy:
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Simple Harmonic OscillatorQuantum harmonic oscillator
Eigenvalues and eigenfunctions
H !n= E
n!n
H !0= E
0!0
We have seen that the states
are energy eigenstates with energy .
Thus starting with the lowest energy E0, the energy eigenvalues are
E0,E
0+ !! ,E
0+ 2!! ,......
En= E
0+ n!! = n +
1
2
"#$
%&'!!
E0=!!
2
a† !
n
En+ !!( )
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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
!"#
$%&!'
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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
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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)
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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
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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.
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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.