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Solving Schrodinger Equation
• If V(x,t)=v(x) than can separate variables
2
2m
2 ( x ,t )
x 2
V ( x ) i
t
assume ( x, t ) ( x ) (t )
2
2m
d 2
dx 2
2 d 2
2 mdx
2
V ( x ) (t ) ( x ) i
V
1
id
dt
G
G is separation constant valid any x or t
Gives 2 ordinary diff. Eqns.
P460 - Sch. wave eqn.
1
d
dt
Solutions to Schrod Eqn
• Gives energy eigenvalues and eigenfunctions (wave functions).
These are quantum states.
• Linear combinations of eigenfunctions are also solutions. For
discrete solutions
( x, t ) c11 c2 2 ......cn n
each
i i e iEi t /
If H Hermitian
i orthogonal
*
i
j dx ij
normalized
2
c
i 1
P460 - Sch. wave eqn.
2
id
dt
G
(t ) e iGt /
G=E if 2 energy states, interference/oscillation
2 d 2
V E
2
2mdx
( x, t ) ( x )e iEt /
1D time
independent
Scrod. Eqn.
Solve: know U(x) and boundary conditions
want mathematically well-behaved. Do not want:
( x)
x
2
x
2
No discontinuities. Usually
except if V=0 or =0
in certain regions
P460 - Sch. wave eqn.
3
Linear Operators
• Operator converts one function into another
Of ( x) f ( x) x 2
d f ( x)
Of ( x)
dx
• an operator is linear if (to see, substitute in a function)
if O[ f1 ( x) f 2 ( x)] Of 1 ( x) Of 2 ( x) linear
ex : O
d
dx
• linear suppositions of eigenfunctions also solution if operator is
linear……use “Linear algebra” concepts. Often use linear algebra to
solve non-linear functions….
P460 - Sch. wave eqn.
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Solutions to Schrod Eqn
• Depending on conditions, can have either discrete or continuous
solutions or a combination
( x, t )
iEn t /
C
u
(
x
)
e
n n
n
C ( E )u
E
( x )e
iEn t /
dE
• where Cn and C(E) are determined by taking the dot product of
an arbitrary function with the eigenfunctions u. Any function
in the space can be made from linear combinations
P460 - Sch. wave eqn.
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Solutions to Schrod Eqn
• Linear combinations of eigenfunctions are also solutions. Assume two
energies
( x, t ) c11 c2 2
c1 1e iE1t / c2 2 e iE2t /
assume know wave function at t=0
( x,0)
1
2
5
7
2
7
• at later times the state can oscillate between the two states probability to be at any x has a time dependence
| ( x, t ) | | c1 1 ( x) |
2
c1c2 ( 2 e
*
1
i ( E2 E1 ) t /
2
| c2 2 ( x) |
1e
P460 - Sch. wave eqn.
*
2
2
i ( E1 E2 ) t /
6
)
Example 3-1
• Boundary conditions (including the functions being mathematically
well behaved) can cause only certain, discrete eigenfunctions
d f ( )
f ( )
d
with
f ( ) f ( 2 )
i
• solve eigenvalue equation
i
1
d f ( )
d f ( )
eigenvalue or
i d
f ( )
d
f ( )
int egrate ln f ( ) i cons tan t
or
f ( ) f (0)e i
• impose the periodic condition to find the allowed eigenvalues
e i ( 2 ) 1 0,1,2, etc
P460 - Sch. wave eqn.
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Square Well Potential
• Start with the simplest potential
V ( x ) V0
V ( x) 0
| x |
| x |
a
2
V0 finite or
a
2
(" in" the well )
For value
( x) 0 for | x |
a
2
V is finite
Boundary condition is that is continuous:give:
out ( a2 ) in ( a2 ) 0 if V0
V
-a/2
a/2
P460 - Sch. wave eqn.
0
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Infinite Square Well Potential
• Solve S.E. where V=0
2
2 d
2 m dx 2
E
A sin kx, B cos kx, Ceikx
Boundary condition quanitizes k/E, 2 classes
Odd
Even
=Bcos(knx)
=Asin(knx)
kn=n/a
kn=n/a
n=1,3,5...
n=2,4,6...
(x)=(-x)
(x)=-(-x)
En
p2
2m
2k 2
2m
22n 2
2 ma 2
h 2n 2
8 ma 2
as n 0 E min E1 0
P460 - Sch. wave eqn.
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Parity
• Parity operator P
x -x (mirror)
P ( x ) ( x )
• determine eigenvalues
Pu ( x) u ( x)
P 2u ( x) Pu ( x) 2u ( x )
but P[ Pu ( x)] Pu ( x ) u ( x ) 2 1 1
even and odd functions are eigenfunctions of P
Odd : Px x
Even : Px 2 x 2
P sin x sin x
P
P cos x cos x
P
x
2
x 2
x
2
x 2
• any function can be split into even and odd
( x) 12 [ ( x) ( x)] 12 [ ( x) ( x )]
( x) ( x) ( x)
1
2 (1 P )
P ( x ) ( x ) ( x )
12 (1 P )
P460 - Sch. wave eqn.
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Parity
• If V(x) is an even function then H is also even then H and P commute
[ H , P ] HP PH 0
• and parity is a constant. If the initial state is even it stays even, odd
stays odd. Semi-prove:
• time development of a wavefunction is given by
i
H ( x, t )
t
• do the same for P when [H,P]=0
i
( P )
H [ P ( x , t )] P[ H ( x , t )]
t
• and so a state of definite parity (+,-) doesn’t change parity over time;
parity is conserved (strong and EM forces conserve, weak force does
not)
P460 - Sch. wave eqn.
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Infinite Square Well Potential
• Need to normalize the wavefunction. Look up in integral tables
2
|
(
x
)
|
dx
A
a
2
A2 sin
2
nx
a
dx 1
a
2
2/a
What is the minimum energy of an electron confined to a nucleus? Let a = 10-14m
= 10 F
Emin
2 2
2 ma 2
( hc ) 2
8 mc 2 a 2
(1240MeVF ) 2
8.51MeV (10 F ) 2
4000 MeV relativist ic
Emin
m2 p 2
k
hc
2a
redo
m 2 (k ) 2
1240MeV F
210 F
P460 - Sch. wave eqn.
60 MeV
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Infinite Square Well Density of States
• The density of states is an important item in determining the
probability that an interaction or decay will occur
• it is defined as
dn
(E)
• for the infinite well
n number of states
dE
8ma 2
n
E cE
h2
dn
c
1
c
2ndn cdE
dE
2n
2 E
2
• For electron with a = 1mm, what is the number of states within 0.0001
eV about 0.01 eV?
8 511000eV (. 1cm ) 2
c
2.7 1012 eV 1
4
2
(1.24 10 eVcm )
dn
1
n
E
dE
2
c
1
E
E
2
P460 - Sch. wave eqn.
2.7 1012
.0001eV 820
.01eV
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Example 3-5
• Particle in box with width a and a wavefunction of
( x ) A( x / a ) 0 x a / 2
( x ) A(1 x / a ) a / 2 x a
A
12 / a
• Find the probability that a measurement of the energy gives the
eigenvalue En
Au
n
n
( x)
un
2
a
sin
nx
a
n
a
An
( x )u
n
a/2
dx 2
0
2
0
24
12 x
a a
2
nx
sin
dx
a
a
1
( 1) n 1
2
n
• With only n=odd only from the symmetry
• The probability to be in state n is then
| An |2
96
Pr ob1 .986
4n 4
Pr ob3
P460 - Sch. wave eqn.
.986
.012
34
14
Free particle wavefunction
• If V=0 everywhere then solutions are
A cos k x , A sin k x , e ikx , e ikx
E
p2
2m
2k 2
2m
• but the exponentials are also eigenfunctions of the momentum
operator
pop i
x
pop ( eikx ) i ik eikx eigenvalue k p
pop ( e ikx ) i ik e ikx k p
• can use to describe left and right traveling waves
• book describes different normalization factors
P460 - Sch. wave eqn.
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