Developing Wave Equation
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Transcript Developing Wave Equation
Solving Schrodinger Equation
• If V(x,t)=v(x) than can separate variables
2
2m
2
( x ,t )
2
x
2
2
d
d
2 mdx
2
dx
2
2
t
( x , t ) ( x ) ( t )
assume
2m
V ( x ) i
2
d
V ( x ) ( t ) ( x ) i
V
1
i d
dt
dt
G
G is separation constant valid any x or t
Gives 2 ordinary diff. Eqns.
P460 - Sch. wave eqn.
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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 ) c1 1 c 2 2 ...... c n n
each
i ie
iE i t /
If H Hermitian
i orthogonal
j dx
*
i
ij
normalized
c
2
i
P460 - Sch. wave eqn.
1
2
i d
G
dt
( t ) e iGt
/
G=E if 2 energy states, interference/oscillation
d
2
2
2 mdx
2
1D time
V E
independent
( x, t ) ( x )e
iEt /
Scrod. Eqn.
Solve: know U(x) and boundary conditions
want mathematically well-behaved. Do not want:
( x)
x
2
x
No discontinuities. Usually
2
except if V=0 or =0
in certain regions
P460 - Sch. wave eqn.
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Linear Operators
• Operator converts one function into another
Of ( x )
f ( x) x
Of ( x )
d f ( x)
2
dx
• an operator is linear if (to see, substitute in a function)
if O [ f 1 ( x ) f 2 ( x )] Of
ex : O
1
( x ) Of
2
( x ) linear
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….
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Solutions to Schrod Eqn
• Depending on conditions, can have either discrete or continuous
solutions or a combination
( x, t )
C nu n ( x )e
iE n t /
n
C ( E )u
E
( x )e
iE n 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 ) c1 1 c 2 2
c1 1 e
iE 1 t /
c 2
e
2
iE 2 t /
assume know wave function at t=0
( x ,0 )
2
7
1
5
7
2
• at later times the state can oscillate between the two states probability to be at any x has a time dependence
| ( x , t ) | | c1
2
c1 c 2 ( 1
*
e
2
1
( x) |
i ( E 2 E1 ) t /
2
| c 2
2
2 1 e
P460 - Sch. wave eqn.
*
( x) |
2
i ( E1 E 2 ) t /
6
)
Example 3-1
• Boundary conditions (including the functions being mathematically
well behaved) can cause only certain, discrete eigenfunctions
i
d f ( )
d
f ( )
f ( )
with
• solve eigenvalue equation
i
1
d f ( )
f ( )
d
int egrate
eigenvalue
ln
f ( )
or
f ( 2 )
or
d f ( )
f ( )
i d
f ( ) i cons tan t
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 |
V0
a
2
a
2
finite
or
(" in " the well )
For value
( x ) 0 for | x |
a
2
V is
finite
Boundary condition is that is continuous:give:
out
(
a
2
)
(
in
a
2
) 0 if V 0
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
2m
d
2
dx
2
E
A sin kx , B cos kx , Ce
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
2
p
2m
2
2
k
2m
as n 0 E min
2
2
n
2 ma
2
2
h
2
n
8 ma
2
2
E1 0
P460 - Sch. wave eqn.
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ikx
Parity
x -x (mirror)
• Parity operator P
P ( x) ( x)
• determine eigenvalues
Pu ( x ) u ( x )
P u ( x) Pu ( x) u ( x)
2
2
but P [ Pu ( x )] Pu ( x ) u ( x )
1 1
2
even and odd functions are eigenfunctions of P
Odd
:
Even :
Px x
Px
x
2
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)
1
2
( x)
[ ( x ) ( x )]
( x)
P ( x)
( x)
(x)
( x)
P460 - Sch. wave eqn.
1
2
[ ( x ) ( x )]
1
2
(1 P )
1
2
(1 P )
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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
t
H ( x, t )
• do the same for P when [H,P]=0
i
(P )
t
H [ P ( x , t )] P [ H ( x , 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
a
2
| ( x) |
2
A
2
sin
nx
a
2
dx 1
a
2
dx
A
2 / a
What is the minimum energy of an electron confined to a nucleus? Let a = 10-14m
= 10 F
E min
4000
E min
2
2
2 ma
2
MeV
m
k
2
hc
2a
( hc )
8 mc
2
2
a
2
( 1240 MeVF
2
2
8 . 51 MeV ( 10 F )
relativist
p
)
m
1240 MeV F
2 10 F
P460 - Sch. wave eqn.
2
ic
2
redo
( k )
2
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 )
n number
of
states
dE
• for the infinite well
n
2
8 ma
h
2 ndn cdE
dn
2
E cE
2
c
dE
2n
1
c
2
E
• For electron with a = 1mm, what is the number of states within 0.0001
eV about 0.01 eV?
c
8 511000 eV (. 1 cm )
n
(1 . 24 10
dn
dE
E
4
eVcm )
1
c
2
E
2
2 . 7 10
2
E
P460 - Sch. wave eqn.
1
2
12
2 . 7 10
eV
1
12
. 0001 eV
820
. 01 eV
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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
12 / a
a / 2 x a
• Find the probability that a measurement of the energy gives the
eigenvalue En
un
An u n ( x )
2
a
sin
nx
a
n
a
An
a /2
( x ) u n dx 2
0
2
12
x
a
a
0
24
1
n
2
( 1)
2
sin
a
nx
dx
a
n 1
• With only n=odd only from the symmetry
• The probability to be in state n is then
| An |
2
96
4n 4
Pr ob 1 . 986
Pr ob 3
P460 - Sch. wave eqn.
. 986
3
4
. 012
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Free particle wavefunction
• If V=0 everywhere then solutions are
A cos kx , A sin kx , e ikx , e ikx
E
p
2
2m
2
k
2m
2
• but the exponentials are also eigenfunctions of the momentum
operator
p op i
p op ( e
p op ( e
ikx
x
) i ik e
ikx
ikx
) i ik e
eigenvalue
ikx
k p
k p
• can use to describe left and right traveling waves
• book describes different normalization factors
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