Schrodinger Wave Equation
11
P460 - Sch. wave e qn. 1 Schrodinger Wave Equation • Schrodinger equation is the first (and easiest) • works for non-relativistic spin- less particles (spin added ad-hoc) • guess at form: conserve energy, well-behaved, predictive, consistent with =h/p • free particle waves K for operator is dx d m m p K p k dx d momentum for operator is dx d i p i ik dx d Ce x ikx 2 2 2 2 2 2 2 2 2 2 2 ) (
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Schrodinger Wave Equation. Schrodinger equation is the first (and easiest) works for non-relativistic spin-less particles (spin added ad-hoc) guess at form: conserve energy, well-behaved, predictive, consistent with l =h/p free particle waves. Schrodinger Wave Equation. - PowerPoint PPT Presentation
Transcript of Schrodinger Wave Equation
P460 - Sch. wave eqn. 1
Schrodinger Wave Equation
• Schrodinger equation is the first (and easiest)
• works for non-relativistic spin-less particles (spin added ad-hoc)
• guess at form: conserve energy, well-behaved, predictive, consistent with =h/p
• free particle waves
Kforoperatorisdx
d
mm
pK
pk
dx
d
momentumforoperatorisdx
di
piik
dx
d
Cex ikx
2
222
2
22
2
2
22
)(
P460 - Sch. wave eqn. 2
Schrodinger Wave Equation• kinetic + potential = “total” energy K + U = E
• with operator form for momentum and K gives
operatorEist
iorit
etxifinspectionby
eqnDSchEUdx
d
mtkxi
)(
2
22
),(
..12
Giving 1D time-dependent SE
titxtxV
dx
d
m
),(),(2 2
22
For 3D:2
2
2
dx
d
P460 - Sch. wave eqn. 3
Operators• Operators transform one function to
another. Some operators have eigenvalues and eigenfunctions
Etieigenvalue
pkxieigenvalue
soandikffx
eigenvalue
efioneigenfunctforsolve
fOfoperatorsaret
andx
tkxi
/:
/:
:
)(
In x-space or t-space let p or E be represented by the
operator whose eigenvalues are p or E
Only some functions are eigenfunctions.
Only some values are eigenvalues
P460 - Sch. wave eqn. 4
Interpret wave function as probability amplitude for being in interval dx
dxt
iE
commutetdoesndxx
ip
commutesdxxx
fofvalueaveffdxf
objectfornormalizedifdx
x
*
*
2
*
2
)'(
)(||
.
11||
P460 - Sch. wave eqn. 5
evenxxelsewhere
ax
aine
a
xAtx iE
)()(.022
cos),( /
No forces. V=0 solve Schr. Eq
Ema
ti
xm
2
22
2
22
2
2
Find average values
0sincos
)(0
*
*
XdxXCdxxi
p
evenoddevendxxx
x
Example
P460 - Sch. wave eqn. 6
57.0
)(
18.)(
033.0)16
(2
cos
2
2
22*
2
22
2
22*2
22
22
2
2
2222222
pxapp
adx
a
dxx
ip
axxxx
aa
dxxAdxxx
rms
x
P460 - Sch. wave eqn. 7
Solving Schrodinger Equation
• If V(x,t)=v(x) than can separate variables
GV
ixtxV
txtxassume
ixV
dtdi
mdx
d
dtd
dx
dm
tx
txm
1
)()()(
)()(),(
)(
2
22
2
22
2
22
2
2
),(2
G is separation constant valid any x or t
Gives 2 ordinary diff. Eqns.
P460 - Sch. wave eqn. 8
/)( iGtdtdi etG
G=E if 2 energy states, interference/oscillation
/
2
22
)(),(
2iEtextx
EVmdx
d
1D time
independent
Scrod. Eqn.
Solve: know U(x) and boundary conditions
want mathematically well-behaved.
2
2
)(
x
x
x
No discontinuities. Usually
except if V=0 or =0
in certain regions
P460 - Sch. wave eqn. 9
Solutions to Schrod Eqn
• Gives energy eigenvalues and eigenfunctions (wave functions). These are quantum states.
• Linear combinationsof eigenfunctions are also solutions
1
......),(
2
*
/
2211
i
ijjii
tiEii
nn
cnormalized
dxorthogonal
eeach
ccctxi
P460 - Sch. wave eqn. 10
Solutions to Schrod Eqn
• Linear combinationsof eigenfunctions are also solutions. Asuume two energies
• assume know wave function at t=0
• at later times the state can oscillate between the two states
/22
/11
2211
21
),(tiEtiE ecec
cctx
275
172)0,( x
)(
||||||/)(
1*2
/)(2
*121
222
211
2
2112 tEEitEEi eecc
cc
P460 - Sch. wave eqn. 11
The normalization of a wave function doesn’t change with time (unless decays). From Griffiths:
dxdxtx ttdtd )(|),(|
**2
Use S.E. to substitute for tt /,/ *
),()(
)(*
2
*2
2
2
*2
*2
2
txJxxxmi
x
xxmi
t
J(x,t) is the probability current. Tells rate at whichprobability is “flowing” past point xsubstitute into integral and evaluate
|)(**
22
xxmi
dtd dx
The wave function must go to 0 at infinity and so this is equal 0
