(1) experimental evidence shows the particles of microscopic systems moves according to the laws of...
TRANSCRIPT
(1) Experimental evidence shows the particles of microscopic systems moves
according to the laws of wave motion, and not according to the Newton laws of
motion obeyed by the particles of macroscopic system.
(2) Planck’s thermal radiation : discrete energy level.
(3) de Broglie wave: connect particle momentum and wavelength by Planck constant
(4) A theory is needed to treat more complicated cases: Schroedinger’s theory of
quantum mechanics.
5.1 Introduction
Chapter 5 Schroedinger theory of quantum mechanics
The Schroedinger equation is a partial differential equation has a
solution .
The equation may include
),( tx
t
tx
x
tx
t
tx
x
tx2
2
2
2 ),(or
),(or
),(or
),(
Ex :
Chapter 5 Schroedinger theory of quantum mechanics
2 and
2for ,)sin()(2sin),( ktkxt
xtx
)sin(),(
,)cos(),(
)sin(),(
,)cos(),(
22
2
22
2
tkxt
txtkx
t
tx
tkxkx
txtkxk
x
tx
5.2 Plausibility argument leading to Schroedinger equation
The reasonable assumption concerning about the wave equations:
(1) de Broglie-Einstein relation:
(2) total energy:
(3) linear wave function :
(4) potential energy :
hEph / and /
mpKVKE 2/ 2
),( tx ),(),(),( 2211 txctxctx
),( txV 0constant /),( FVxtxVF
t
txitxtxV
x
tx
m
k
iim
VmkiVk
i
VkVk
tkxVktkxVk
tkxtkxtx
tkxVtkxktkx
tkxtxVtxVt
txtxtxV
x
tx
t
tx
x
tx
m
k
htxVmk
hmhhmpE
),(),(),(
),(
2
) (take and 2/
2/for
1/0
/ and
0)sin(][)cos(][
)sin()cos(),( as chosen onwavefunctibetter The
)cos()sin()sin(
)sin(),( and ),( If
),(),(),(
),( :equation wave trial
),( termright the ,
),(
2 termleft the
2/for ),(2/
2/2/
2
222
2
022
02
2
02
02
02
02
02
0
2
2
2
222
22
222
Chapter 5 Schroedinger theory of quantum mechanics
Schroedinger wave equation
5.3 Born’s interpretation of wave functions
),(),(),(
)sin()cos(),(
),(),(
),(
2
*
02
22
txtxtxP
tkxitkxtxt
txitxV
x
tx
m
Max Born (1926):
complex wave function
probability density
P(x,t)dx is the probability that the particle with wave function Ψ(x,t) will
be found at a coordinate between x and x+dx.
Chapter 5 Schroedinger theory of quantum mechanics
Classical wave theory:
Wave function is a real function. real is c )sin()cos(),(
),(1),(2
2
22
2
tkxctkxtxt
tx
cx
tx
Chapter 5 Schroedinger theory of quantum mechanics
Ex: (1) Evaluate the probability density for the simple harmonic oscillator
lowest energy state wave function
(2) Evaluate the probability density of S.H.O. in classical mechanics.
tmCixCm eAetx )/)(2/()2/( 2
),(
/2at
2//22
2
2
1
2
1
/
:mechanics Classical (2)
)0(at
),(),(
:mechanics Quantum (1)
max
2
22
22
2
2max
)/(2* 2
CExP
cxEm
BP
cxE
mv
Cx mvVKE
vBP
xAP
eAtxtxP xCm
Q.M.
C.M.
In C.M., no uncertainty principle is an error.
tmCixCm eAetx /)2/()2/( 2
),(
Ex: Normalize the wave function of S.H.O. expressed as
Chapter 5 Schroedinger theory of quantum mechanics
4/18/1
4/12/1
0
)/(
0
)/(2
)/(2*
)/()(
)(2/)(
12
1
2
2
2
CmA
Cmdxe
dxeA
dxeAdxPdx
xCm
xCm
xCm
aI
ade dde
dxdyedyedxeI
dxeI
aπ/ a
yxaayax
ax
/2
142
0
2
0 0
0 0
)(
00
2
0
22
2222
2
Chapter 5 Schroedinger theory of quantum mechanics
5.4 Expectation values
dxtxtxVtxtxV
dxtxxftxxf
dxtxxtxx
txdxtxxtx
dxtxtxdxtxxtx
dxtxPdxtxxPxx
),(),(),(),(
),()(),()(
),(),(
normalized is ),( if ),(),(
),(),(/),(),(
),(/),(
:ist instant theat
particle the of coordinatex the of value expection The
*
*
2*2
*
**
Chapter 5 Schroedinger theory of quantum mechanics
Momentum and Energy operators:
),(),(),()],(2
[),()],(2
[
),(),(
),()/( )]sin()[cos(),(
),(),(
2for ),()]sin()[cos(
),(
)sin()cos(),(
),(),(
),(
2 If
2
2
22
02
22
txEtxt
itxtxVm
ptxtxV
xm
-
tiE
t
txitxE
txEitkxitkxit
txx
ipx
txitxp
pktx
pitkxitkxik
x
tx
tkxitkxtxt
txitxV
x
tx
m
-
opop
op
op
Chapter 5 Schroedinger theory of quantum mechanics
dxx
txtxip
dxx
txtxip
txtxidxtxtxx
ip
),(),(
nonsense),(),( (2)
0)],(),([)],(),([(1)
*
*
**
Momentum expectation value
Energy expectation value
dxtxtx
ixftxtpxf
dxtxtxVxm
tx
dxt
txtxiE
op ),(),,(),(),,(
),()],(2
)[,(
),(),(
*
2
22*
*
Ex: Consider a particle of mass m which can move freely along the x axis between two walls at x=-a/2 and x=+a/2, and the particle can not penetrate the two walls. Try to find the wave function of the particle and energy.
Chapter 5 Schroedinger theory of quantum mechanics
2 ,2/for ),( and 2/2/for 0),( a/xaxtxVaxatxV
.....5,3,1 )cos()( 22
odd is n 0)2/cos()2/( and )(c(x)
)()(2)(
)()(
2 (1)
)(
)(
1)(
)(
1
2)()(
1
)()(
)()(
2
0)2/()2( and )()(),(set
),(),(
2
2
22222
222
2
2
2
2
22
2
22
2
22
na
xnx
ma
n
m
k
nkakaakxos
xkxm
dx
xdx
dx
xd
m
dt
tdT
tTi
dx
xd
xmtTx
dt
tdTxAi
dx
xdtAT
m
a,ta/tTxAtxt
txi
x
tx
m
Chapter 5 Schroedinger theory of quantum mechanics
/12
22
1
/2
222
*
/2
2/
2/
22*
/
/
1)cos(2
),( 2
1 state ground
: :),(
531 )cos(2
),(2
level energy
)cos(2),(
),(
)cos(2
),(/2/2||
)(cos||1),(),(
)cos(),(
)()()(
(2)
tiE
nn
iEtnn
ti
a
a
ti
ti
ea
x
atx
maEn
Etx
....,,nea
xn
atx
ma
nE
dxa
xn
adx
t
txitxE
ea
xn
atxaAaA
dxa
xnAtxtx
ea
xnAtx
etTtTdt
tdTi
eigenfunction eigenvalue
Vm
pVKH
txEtxH
op
2
),(),(2
Chapter 5 Schroedinger theory of quantum mechanics
Eigenvalue equation
Hamiltonian or total energy operator
22
1
12
and 2
1
12
2)(
)),(
)(,(
]2
1
12[)(cos
2
0)sin()cos(),())(,(
0)(cos2
),(),(
22
2222
22222
2
222
2
22*2
2
2
2
222/
2/
22
2/
2/
*
2/
2/
2*
npx
a
nppp
n
n
a
xxxxxxxxx
a
ndx
x
txtxp
n
adx
a
xnx
ax
dxa
xn
a
xntx
xitxp
dxa
xnx
adxtxxtxx
a
a
a
a
a
a
Uncertainty principle
5.5 The time-independent Schroedinger equation
Chapter 5 Schroedinger theory of quantum mechanics
/
2
22
/
2
22
2
22
2
22
)()(
:)(
)()()()(
2
energy total is /for
)2sin()2cos()()()(
)(
)(
1)()(
)(
)(
1
2)()(
1
)()()()()(
)()(
2
(1) eq. intoput )()(),(set )(),( if
-(1)--- ),(
),(),(),(
2
iEt
iEt
exx,t
x
xExxVdx
xd
m
hEhE
th
Eit
h
EettE
dt
tdi
Edt
td
tixxV
dx
xd
xmtx
dt
tdxitxxV
dx
xdt
m
txtxxVtxVt
txitxtxV
x
tx
m
time-independent Schroedinger equation
eigenfunction
wave function
Chapter 5 Schroedinger theory of quantum mechanics
5.6 Required properties of eigenfunctions
must be finite must be finite
must be single valued must be single valued
must be continuous must be continuous
dxxdx /)( )( dxxdx /)( )(
dxxdx /)( )(
(1) Physical measurable quantities, e.g., p, x, are all finite and single-valued, so are finite and single-valued.
(2) is finite, it is necessary is continuous.
(3) For finite V(x), E and , must be continuous.
dxxdx /)( ,)(
/)( dxxd )(x)(x 22 /)( dxxd
Chapter 5 Schroedinger theory of quantum mechanics
Ex: When a particle is in a state such that a measurement of its total energy
can lead (1) only to a single result, the eigenvalue E, it is described by the
wave function (2) two results, the eigenvalue
wave function is
What are their probability density?
/)(),( iEtextx 21 , EE /
22/
1121 )()(),( tiEtiE excexctx
2
)()(
)()(c
)()(cc)()(c
])()([])()([ (2)
)()()()( (1)
1212
/)(2
*12
*1
/)(1
*21
*2
2*22
*21
*11
*1
/22
/11
/*2
*2
/*1
*1
*
*//**
12
12
2121
h
EEEE
exxcc
exxc
xxxxc
excexcexcexc
xxexex
tEEi
tEEi
tiEtiEtiEtiE
iEtiEt
oscillating frequency of probability density
independent of time