Download Quantum Physics 2005 Notes-4 The Schrodinger Equation (Chapters 6 + 7)

Quantum Physics 2005
Notes-4
The Schrodinger Equation
(Chapters 6 + 7)
Notes 4
Quantum Physics F2005
1
The Schrodinger Equation
Hˆ ! ! Eˆ!
2
2
h "
"!
#
! " V ( x, t )! ! ih
2
"t
2m "x
• We can use the relationship between time
and space derivatives to explore the behavior
of the wavefunctions.
Notes 4
Quantum Physics F2005
2
Moving particles and probability flow
2

"P ( x, t ) "! *!
"! *
"
!
h
"
!
*
*
!
!!
"!
!!  #
2
"t
"t
"t
"t
2
im
"
x


 h " 2! * 
 " !  2im "x 2 



h  * " 2!
" 2! * 
"  h  * "! "! *  
!#
#!
!# 
#
! 
!
!
2
2 
"x
"x 
"x  2im 
"x
"x  
2im 
h  * "! "! * 
!
!  , we have,
Defining, j ( x, t ) !
#

2im 
"x
"x 
"P ( x, t )
"j ( x, t )
!#
"t
"x
where j ( x, t ) is called the probability current density.
Notes 4
Quantum Physics F2005
3
Interpreting probability flow
• We can understand this equation if we think
about probability in a finite region and current
flow into and out of this region.
"
"b
P(# a, b $ , t ) ! ∫ P( x, t )dx ! j (a, t ) # j (b, t )
"t
"t a
This is a conservation law that states that the only way
the probability of finding a particle in a region can
change is if the probability of finding it outside the
region increases accordingly..
Notes 4
Quantum Physics F2005
4
Probability current for a pure momentum state
Taking $ p 0 ( x, t ) ! Aei ( p0 x # E0t ) / h
h  * "! "! * 
!
j ( x, t ) !
#
!
2im 
"x
"x 
2
hA
!
2im
2
 # i ( p0 x # E0t ) / h ip0 i ( p0 x # E0t ) / h  #ip0  # i ( p0 x # E0t ) / h i ( p0 x # E0t ) / h 
e
#
e
e
e

h
h




2
h A 2ip0
A
2
!
!
p0 ! A v0
2im h
m
probability current is constant in space,
therefore what flows into a volume must flow out.
Notes 4
Quantum Physics F2005
5
Probability flow in three dimensions
We are interested in the probability of finding
the particle in a finite volume V.
r
PV ! ∫∫∫ P(r , t )d%
r r
#i h * r r r
r r * r
$ (r , t )&$ (r , t ) # $ ( r , t )&$ ( r , t ) , so
We define j (r , t ) !
2m
"P r
" &j ! 0
"t
r r
"
r
∫∫∫ V P(r , t )d% " ∫∫ j ' da ! 0
"t
A
%
Notes 4
&
Quantum Physics F2005
6
Time evolution of a wave packet -1
• A wave packet is a linear combination of an
infinite number of extended singlewavelength with infinitesimally differing
wavenumbers.
1 )
i % kx #'t &
$ ( x, t ) !
A
k
e
dk
(
)
∫
2( #)
1 )
# i % kx &
dk
where A(k ) !
∫ $ ( x, 0)e
2( #)
• This definition allows us to predict the time
evolution of the wave packet.
Notes 4
Quantum Physics F2005
7
Time evolution of a wave packet – 2
Gaussian example
Assuming the spatial part of the wavefunction at t ! 0 is a
Normal Gaussian function centered at 0 and moving with
average wavenumber k0 ;
1/ 2
 1 
$ ( x, 0) ! 

 * x 2( 
e
ik0 x # x 2 / 4* x2
e
.
1/ 2
1
So A(k ) !
2(
 1  ) i ( k0 # k ) x # x2 / 4* x2
e
dx

 ∫e
 * x 2(  #)
(We have already solved for A(k) in Notes 2, page 33. )
1
4
 2 2  #* x2 ( k # k0 )2
A(k ) !  * x  e
(

Notes 4
Quantum Physics F2005
8
Time evolution of a wave packet – 3
Gaussian example
• Now, since we know A(k), we can solve for the time evolution of
$(x,t).
i
1 )
A
k
e
(
)
$ ( x, t ) !
∫
2( #)
2
x
3
kx # k 2 t / 2 m
1
4 )
 * x2 
#* x2 ( k # k0 ) 2 i
dk !  3  ∫ e
e
 2(  #)
kx # k 2t / 2 m
dk
1
4
 *  i k0 x # k02t / 2 m )
  2
h  2  hk 0
 
!
#
"
#
#
e
i
t
u
i
t
ix
exp
*
∫
 x


 u  du


2m 
#)
 m
 
 
 2( 
which is an integral of a standard form, ∫ exp(#+ u 2 # , u )du.
1
4
 * x2  i
$ ( x, t ) !  3  e
 2( 
Notes 4
k0 x # k02 t / 2 m
hk 0  
 
x
t 
#
#
 
(
m 
' exp  

h
h

 4 * 2 " i
* x2 " i
t
t

  x
2m
2m  
Quantum Physics F2005
9
Time evolution of a wave packet – 4
Gaussian example
• An animation of a travelling quantum wave packet can be found
at: travelling_gaussian.mws
• Notes:
• Look at the probability distribution. You can see the motion
of the center of the wavepacket is the same for whatever
initial width you choose. The speed only depends on the
initial choice of k0.
• When the initial packet is narrower, the packet spreads more
quickly. You can see why this is by looking at the real part of
the wave packet (where you can see the waves.)
• Short wave components travel faster than longer wave
components, as you can see in the display of the real
part of psi.
• This effect can be observed in the transport of light pulses
down fiber optics.
Notes 4
Quantum Physics F2005
10
Thus far in our story,
• We have shown some properties of wave
functions and state functions.
– Some rules governing the form of the wave
functions and some plausible wave functions.
– Given a wave function to describe a particle, how
do we deduce observable properties?
– What is a plausible wave equation?
• We will now start finding solutions to the wave
equation under various well-studied
conditions.
Notes 4
Quantum Physics F2005
11
Things you should understand
and be able to do*
• Understand and do simple calculations related to the important
experiments listed in Notes 1 (Photoelectric, Compton,
Diffraction)
• Understand and use the Einstein and DeBroglie equations
relating particle properties (kinetic energy and momentum) to
wave properties (frequency and wavelength) (p. 39 in Morrison).
• Understand and perform simple calculations using the
Heidenberg Uncertainty Principle (p. 9, Morrison)
• Do calculations with probability distributions (normalization,
expectation value, variance, and standard deviation). You
should also be able to look at a graph of probability and make a
reasonable guess of average position and standard deviation.
*(on a quiz, for example)
Notes 4
Quantum Physics F2005
12
Things you should understand
and be able to do –2
• Understand the concept of the complex state function and its
relation to probability, especially the wave function and the
probability of finding the particle somewhere.
• Know the physical limitations on the form of wave functions. (p.
78)
• Recognize a pure momentum state (single wavelength,
travelling wave) and pick out the wavelength, wave number,
frequency, period, and phase velocity. (pp. 107 ff)
• Transform back and forth between the wave function (real
space) representation and the momentum amplitude function
(momentum space) representation of a quantum state.
– Set up a transform in each direction.
– Perform the math for some simpler cases. Use symmetry to
simplify or avoid calculation.
– Have a rudimentary understanding of how the real space wave
function is related to the amplitude function.
• especially, the Heisenberg
Uncertainty Principle
Notes 4
Quantum Physics F2005
13
Things you should understand
and be able to do -3
• Understand the concepts of the wave packet and
group velocity. (p. 113)
• Be able to perform simple operator arithmatic.(p. 159)
• Know and love useful operators for important
observables in real space (p. 173) and momentum
space (position, momentum, kinetic energy, total
energy)
• Know the Schrodinger equation and how one
simplifies it into the time independent Schrodinger
equation.
Notes 4
Quantum Physics F2005
14
Another motivational interlude
• Many of the really amazing apparent paradoxes of
physical understanding have roots in quantum
physics. (A particle interfering with itself is odd. A
particle being everywhere at once. Quantum
cryptography.)
• The understanding of many of the most common
things is based in quantum physics. (Light emission
by a fluorescent bulb. Lasers. Semiconductors. The
folding of proteins. Photosynthesis. Chemical
processes.)
• Developing new nanotechnology requires an
understanding of quantum physics.
Notes 4
Quantum Physics F2005
15
A special set of solutions –
Constant V
The full TDSE:
"$ ( x, t )
h2 " 2
#
$ ( x, t ) " V ( x, t ) $ ( x, t ) ! i h
2
2m "x
"t
Let's assume that V does not change with time:
"$ ( x, t )
h2 " 2
#
$ ( x, t ) " V ( x ) $ ( x, t ) ! i h
2
2m "x
"t
Now we'll search for solutions using the
separation of variables method:
$ ( x, t ) ! ! ( x)% (t )
Notes 4
Quantum Physics F2005
16
Constant-V solutions
"! ( x )% (t )
h 2 " 2 (! ( x)% (t ))
#
" V ( x)! ( x)% (t ) ! ih
2
2m
"x
"t
" 2! ( x)
"% (t )
h2
#
"
(
)
(
)
(
)
!
(
)
V
x
x
t
x
i
% (t )
!
%
!
h
2m
"x 2
"t
Divide both sides by $ ( x, t )
"% (t )
! ( x)
h 2 % (t ) " 2! ( x)
#
" V ( x) !
ih
2
2m ! ( x)% (t ) "x
"t
! ( x)% (t )
Now we see that the left is only a fn of x and the right only t.
The only way this equation can be true for all x and t is if
each side is equal to the same constant.
Notes 4
Quantum Physics F2005
17
Constant-V solutions
So now we have two separated equations:
h 2 1 " 2! ( x)
#
" V ( x) ! +
2
2m ! ( x) "x
1
"% (t )
ih
!+
% (t )
"t
Looking at the t equation:
"% (t )
ih
# +% (t ) ! 0
"t
yields % (t ) ! % (0)e # i+ t / h
and by association with the Einstein-DeBroglie relations:
+ ! h' ! E
Notes 4
Quantum Physics F2005
18
Constant-V solutions
So now we have:
$ ( x, t ) ! ! ( x)% (t ) ! ! ( x)e # iEt / h
Where the total energy is a constant
The new equation in x,
of the particle.
h 2 " 2! ( x)
#
" V ( x)! ( x) ! E! ( x)
2
2m "x
is called the time-independent
Schrodinger Equation
The wave function solutions to the timeindependent SE are called stationary states
because the energy is constant
Notes 4
Quantum Physics F2005
19
Special solutions for the onedimensional Schrodinger equation
with constant potential:
Stationary States
Notes 4
Quantum Physics F2005
20
An unconfined particle:
Constant potential – V=0 everywhere
Notes 4
Quantum Physics F2005
21
Free Space: V= same everywhere
Stationary states of the free particle wave -1
" 2! ( x) 2m( E # V )
! ( x) ! 0
"
2
2
h
"x
should look familiar from your diff eq course.
 "2 y

2
or Schaum's Outline  2 " k y ! 0  .
 "x

The solutions are
sin kx, cos kx, eikx , or e # ikx
2m( E # V )
where k !
h2
By convention, we set V ! 0 for free space.
Notes 4
Quantum Physics F2005
22
The general solution vs the specific case
The free particle wave -2
• There are an infinite number of possible solutions to
the free space Schrodinger equation. All we have
found is the relation between the possible time
solutions and the possible space solutions.
• We need to give more information about the state for
you to limit the set of possible solutions.
– If we specify the energy, E, then the set of possible k’s is
limited to two possibilities (+ and -), but this still leaves us
with sine and cosine, or +k, -k solutions.
– We need to be given other limitations, such as the value of
the wavefunction at certain points/ certain times.
• We will spend a significant portion of this course
solving special cases, given interesting V(x) and
boundary conditions and/or initial conditions.
Notes 4
Quantum Physics F2005
23
Wavefunction for a free particle wave –3
The spatial part of the wave function is a linear
combination of a complete set of solutions.
Choosing the exponential function representation,
! ( x) ! aeikx " be # ikx so,
$ ( x, t ) ! ! ( x)% (t ) ! % aeikx " be# ikx & eiEt / h ! aei ( kx "'t ) " be # i ( kx #'t )
If we specific the direction of propagation to be +x, then
$ ( x, t ) ! ! ( x)% (t ) ! be # ikx eiEt / h ! be# i ( kx #'t ) .
Looking at the probability density: $ * $ ! b 2 ! constant
)
2
)
Let's try normalizing this: 1= ∫ $ * $dx ! b ∫ dx
-)
-)
This means we have to let b ⇒ 0 as x ⇒ ).
Notes 4
Quantum Physics F2005
24
Wavefunction for a free particle wave –4
momentum
Since the position is completely indeterminate, let's see what the
momentum is:
)
)
"
*
2
i ( kx #'t )
# i ( kx #'t )
$
$
#
i
dx
b
e
i
(
ik
)
e
dx
h
h
∫
∫
"x
p ! #)
! #)
! hk
)
*
2
∫ $ $dx
b ∫ dx
#)
#)
2
)
"2
# i ( kx #'t )
2
2 "
i ( kx #'t )
$
#
dx
dx
b
e
e
h
∫ $ #h
∫
2
2
2 2
"x
"x
! #) )
! #)
k
!
h
)
2
*
$
$
b
dx
∫
∫ dx
)
p2
)
*
2
#)
#)
So,
p
2
# p
2
2
2
2
! h k # % hk & ! 0
This state has a perfectly defined momentum.
Notes 4
Quantum Physics F2005
25
Wavefunction for a free particle – 5
Probability current
j ( x, t ) -
#ih  * "
" *
$
$
#
$
$ 

2m  "x
"x 
#ihb 2 i ( kx #'t )
e
!
(#ik )e # i ( kx #'t ) # e# i ( kx #'t )ike# i ( kx #'t ) 
2m
2 hk
2 p
!b
!b
! Pv
m
m
where P is the probability density
Notes 4
Quantum Physics F2005
26
A particle confined
to a region of space:
The Particle in a Box
with Infinitely High Walls
Notes 4
Quantum Physics F2005
27
Stationary states of a particle in a box -1
Outside the box, $ ( x, t )=0.
Inside the box, V=0, so:
$ ( x, t ) ! # a cos(kx) " b sin(kx) $ eiEt / h
V=0
V=)
x=0
V=)
x=L
or
$ ( x, t ) ! ceikx " de # ikx  eiEt / h
(we are free to choose which one
to use)
Boundary conditions: ! (x . 0)=0;
Notes 4
! (x / L)=0
Note that I have chosen to place the boundaries at 0 and L,
rather than +/- L/2 as we did previously. I have done this to provide
a slightly different approach and solutions, but the basic result is the same
Quantum Physics F2005
28
Particle in a box - 2
Starting with the x=0 boundary condition:
! (0) ! a sin 0 " b cos 0 ! 0
The only way this can be true is if b=0.
Now for the x=L boundary condition:
! ( L) ! a sin kL ! 0
This can be true if a=0 (no wave function at all)
or if kL ! n( .
! n ( x) ! a sin % kn x & and normalizing gives:
! n ( x) !
Notes 4
2
sin % kn x &
L
Quantum Physics F2005
29
Particle in a box - 3
• The imposition of the two boundary
conditions that confine the wavefunction to a
region of space leads to quantization of the
stationary state particle energies.
2
 n( 
! n ( x) !
sin 
x
L
 L 
Each stationary state has a unique energy:
2 2
h 2 kn2
h
(
2
En !
!n
2m
2mL2
Notes 4
Quantum Physics F2005
30
Particle in a box - 4
• If we chose to place the boundaries of the
box at (L/2, we would have found spatial
wave functions like those in the book:
! n( e ) ( x) !
!
(o)
n
( x) !
2
 n(
cos 
L
 L

x  for n=odd

2
 n(
sin 
L
 L

x  for n=even

• The corresponding energy level for a given n
is the same as we found for the 0,L box.
Notes 4
Quantum Physics F2005
31
Particle in a box – 5
Wave functions for stationary states
• Note that however we choose to specify the box, the states have
definite even or odd parity about the center of symmetry of the
potential.
• The existence of this symmetry in the potential led to simple
solutions for the wavefunctions.
!
!
P
P
Notes 4
!
!
P
P
Quantum Physics F2005
32
Particle in a box – 6
Momentum distribution for stationary states
• In a previous homework (and in Notes 3), we
deduced the momentum amplitude functions for
these wavefunctions.
  n( kL 
 n( kL  
sin 
#  sin 
" 

1
2
2 
2 
 2
#
A(k ) !
 
 for even n
 n(
 
L(   n( # k 
k
"


 
  L

 L
 
P(k)
Notes 4
Quantum Physics F2005
33
Particle in a box – 7
Expectation values for stationary states
By observing symmetries and from our previous work on
particle in a box wave functions, we know:
Expectation value of position: xn ! the center of the box
1
1
# 2 2
Uncertainty in position: x ! L
12 2n (
Expectation value of momentum: p ! 0
2
n
Uncertainty in momentum:
Notes 4
2
n
p
n( h
!
L
Quantum Physics F2005
34
Particle in a box –8
Expectation value and uncertainty in energy
2
ˆ
p
The kinetic energy operator is: Tˆ !
.
2m
Since we previously solved for pn2 we can easily solve for T .
pn2
2
1  n( h 
Tn !
!


2 m 2m  L 
The lowest allowed state has n=1,
so expectation value of the kinetic energy is always > zero.
Another remarkable observation is that:
T
2
# T
2
! 0.
There is no uncertainty in the energy!
Notes 4
Quantum Physics F2005
35
Particle in a box – 9
Probability current
2
 n(
sin 
L
 L

x  e # i'nt

#i h  * "
" *
j ( x, t ) $
$
#
$
$ 

2m  "x
"x 
#ih 2
!
# k sin kn x sin knx # k sin kn x sin knx $ ! 0
2m L
everywhere in the box.
$ n ( x) !
Notes 4
Quantum Physics F2005
36
A problem with this example
• A small monster in the closet here is that these
wavefunctions do not satisfy all of the physical
limitations on wavefunctions that we specified earlier
– continuous? – yes
– normalizable? – yes
– continuous derivative? – no, there is a kink at the
boundaries
• This situation has arisen because we have actually
specified an infinite potential energy discontinuity at
the boundaries.
Notes 4
Quantum Physics F2005
37
A general observation about energy levels
• We have found for the particle in the box that
the energy is quantized – it can only take on
certain discrete values that are related to the
dimensions of the box and the mass of the
particle.
• We found no such quantization for the free
particle plabe wave.
• It is a general observation that situations in
which particles are bound give rise to
quantized energies whereas unbound
systems lead to an energy continuum.
Notes 4
Quantum Physics F2005
38
Another observation about energy
The Time Independent Schrodinger Equation
for a free particle also leads to another useful observation:
h2 " 2
Inspecting #
! ! E! we see that the left hand side
2
2m "x
is a measure of the curvature of the wavefunction, whereas the
right hand side is the energy.
Basically, the curvier the wave function, the more energy is has.
This is useful of you want to get an idea of what the
wavefunction might look like for a special potential shape.
Notes 4
Quantum Physics F2005
39
An addition to the language:
Eigenvalue equations, Eigenfunctions
Eigenvalues
The Time Independent Schrodinger equation is a
second-order, ordinary differential equation in x.
It is linear and homogeneous and has 2 independent solutions.
Hˆ ! ! E!
is known in mathematics as an eigenvalue equation because
the operator returns a constant times the original function.
! is called the eigenfunction of the operator Hˆ
E is the corresponding eigenvalue for H!
Notes 4
Quantum Physics F2005
40
The next set of notes will deal with
several specific cases
Step Potential
Barrier Potential
Harmonic Well Potential
Particle in a 3D Box (Quantum Dot)
Notes 4
Quantum Physics F2005
41
Notes 4
Quantum Physics F2005
42