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Chapter 40
Quantum Mechanics
Using Schrödinger Equation
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To study the “particle in a box”
To consider and construct wave functions
To study a finite potential well
To examine quantum mechanical behavior
around a barrier and “tunneling”
To consider the harmonic oscillator—our first
model for molecular vibrations
To study three-dimensional systems
Erwin Schrödinger
1887 – 1961
American physicist
Best known as one of
the creators of quantum
mechanics
His approach was shown
to be equivalent to
Heisenberg’s
Also worked with:
statistical mechanics
color vision
general relativity
Schrödinger Equation
The Schrödinger equation as it applies to a
particle of mass m confined to moving
along the x axis and interacting with its
environment through a potential energy
function U(x) is
h 2 d 2ψ
−
+ Uψ = Eψ
2
2m dx
This is called the time-independent
Schrödinger equation
Schrödinger Equation, cont.
For most cases, the first term in
the Schrödinger equation reduces
to the kinetic energy of the particle
multiplied by the wave function
Solutions to the Schrödinger
equation in different regions must
join smoothly at the boundaries
Schrödinger Equation, final
ψ(x) must be continuous
dψ/dx must also be continuous for
finite values of the potential
energy
Solutions of the Schrödinger Equation
Solutions of the Schrödinger equation may
be very difficult
The Schrödinger equation has been
extremely successful in explaining the
behavior of atomic and nuclear systems
Classical physics failed to explain this behavior
When quantum mechanics is applied to
macroscopic objects, the results agree with
classical physics
Wave Function
The complete wave function ψ for a
system depends on the positions of all
the particles in the system and on time
The function can be written as
r r r
r
r − iωt
Ψ = ( r1 + r2 + r3 + K + rj K,t ) = Ψ ( rj ) e
rj is the position of the jth particle in the
system
ω = 2πƒ is the angular frequency
Wave Function, cont.
The wave function is often complex-valued
The absolute square |ψ|2 = ψ*ψ is always
real and positive
ψ* is the complete conjugate of ψ
It is proportional to the probability per
unit volume of finding a particle at a
given point at some instant
The wave function contains within it all the
information that can be known about the
particle
Wave Function Interpretation
– Single Particle
Ψ cannot be measured
|Ψ|2 is real and can be measured
|Ψ|2 is also called the probability density
The relative probability per unit volume that the
particle will be found at any given point in the
volume
If dV is a small volume element surrounding
some point, the probability of finding the
particle in that volume element is
P(x, y, z) dV = |Ψ |2 dV
Wave Function of a Free Particle
The wave function of a free particle moving
along the x-axis can be written as
ψ(x) = Aeikx
A is the constant amplitude
k = 2π/λ is the angular wave number of the wave
representing the particle
Although the wave function is often
associated with the particle, it is more
properly determined by the particle and its
interaction with its environment
Think of the system wave function instead of the particle
wave function
Wave Function of a Free
Particle, cont.
In general, the probability of
finding the particle in a volume
dV is |ψ|2 dV
With one
- dimensional analysis,
this becomes |ψ|2 dx
The probability of finding the
particle in the arbitrary interval
a ≤ x ≤ b is
b
2
Pab = ∫ ψ dx
a
and is the area under the
curve
Wave Function of a Free
Particle, Final
Because the particle must be somewhere
along the x axis, the sum of all the
probabilities over all values of x must be 1
∞
2
Pab = ∫ ψ dx = 1
−∞
Any wave function satisfying this equation is
said to be normalized
Normalization is simply a statement that the
particle exists at some point in space
Expectation Values
Measurable quantities of a particle can be derived
from ψ
Remember, ψ is not a measurable quantity
Once the wave function is known, it is possible to
calculate the average position you would expect to
find the particle after many measurements
The average position is called the expectation value
of x and is defined as
∞
x ≡ ∫ ψ * xψdx
−∞
Expectation Values, cont.
The expectation value of any
function of x can also be found
∞
f ( x ) = ∫ ψ * f ( x ) ψdx
−∞
The expectation values are analogous
to weighted averages
Such as velocity, momentum, rootmean-square displacement…etc
Summary of Mathematical
Features of a Wave Function
ψ(x) may be a complex function or a
real function, depending on the system
ψ(x) must be defined at all points in
space and be single-valued
ψ(x) must be normalized
ψ(x) must be continuous in space
There must be no discontinuous jumps in
the value of the wave function at any point
Particle in a Box
A particle is confined
to a one-dimensional
region of space
The “box” is onedimensional
The particle is
bouncing elastically
back and forth
between two
impenetrable walls
separated by L
Potential Energy for
a Particle in a Box
As long as the particle is
inside the box, the
potential energy does not
depend on its location
We can choose this energy
value to be zero
The energy is infinitely
large if the particle is
outside the box
This ensures that the wave
function is zero outside the
box
Wave Function for the
Particle in a Box
Since the walls are impenetrable,
there is zero probability of finding
the particle outside the box
ψ(x) = 0 for x < 0 and x > L
The wave function must also be 0
at the walls
The function must be continuous
ψ(0) = 0 and ψ(L) = 0
Schrödinger Equation Applied
to a Particle in a Box
In the region 0 < x < L, where U = 0, the
Schrödinger equation can be expressed in
the form
d 2ψ
2mE
2mE
2
= − 2 ψ = −k ψ where k =
2
dx
h
h
The most general solution to the equation is
ψ(x) = A sin kx + B cos kx
A and B are constants determined by the
boundary and normalization conditions
Particle in a Box (cont’d)
Apply boundary conditions to
ψ(x) = A sin kx + B cos kx
At x = 0, Ψ(0)= 0 + B = 0
→ B=0
At x = L, Ψ(L)= A sin(kL) = 0
→ k = nπ/L
Math – particle in a box
Ψ(x) = A sin(kx), where k = nπ/L
Normalization required
1=
∞
L
−∞
0
2
2
*
dx
A
sin
( kx ) dx
Ψ
Ψ
=
∫
∫
L
L
1
L
∫0 sin (kx )dx = ∫0 2[1 − cos(2kx )]dx = 2
2
So A = √(2/L) and Ψ ( x ) =
2
nπ
sin(
x)
L
L
Graphical Representations for
a Particle in a Box
Wave Function of the Particle
in a Box, cont.
Only certain wavelengths for the
particle are allowed
|ψ|2 is zero at the boundaries
|ψ|2 is zero at other locations as
well, depending on the values of n
The number of zero points
increases by one each time the
quantum number increases by one
Momentum of the Particle
in a Box
Remember the wavelengths are
restricted to specific values
λ= 2 L / n
Therefore, the momentum values
are also restricted
h nh
p= =
λ 2L
Energy of a Particle in a Box
We chose the potential energy of the
particle to be zero inside the box
Therefore, the energy of the particle is
just its kinetic energy (E = ћ2k2/2m,
k=nπ/L)
 h2  2
En = 
n
2 
 8mL 
n = 1, 2, 3 ,K
The energy of the particle is quantized
Energy Level Diagram –
Particle in a Box
The lowest allowed
energy corresponds
to the ground state
En = n2E1 are called
excited states
E = 0 is not an
allowed state
The particle can never
be at rest
Boundary Conditions
Boundary conditions are applied to
determine the allowed states of the system
In the model of a particle under boundary
conditions, an interaction of a particle with
its environment represents one or more
boundary conditions and, if the interaction
restricts the particle to a finite region of
space, results in quantization of the energy
of the system
In general, boundary conditions are related
to the coordinates describing the problem
Potential Wells
A potential well is a graphical
representation of energy
The well is the upward-facing region
of the curve in a potential energy
diagram
The particle in a box is sometimes
said to be in a square well
Due to the shape of the potential energy
diagram
Finite Potential Well
A finite potential well
is pictured
The energy is zero
when the particle is
0<x<L
In region II
The energy has a
finite value outside
this region
Regions I and III
Classical vs. Quantum
Interpretation
According to Classical Mechanics
If the total energy E of the system is less than U, the
particle is permanently bound in the potential well
If the particle were outside the well, its kinetic
energy would be negative (Impossible!!)
According to Quantum Mechanics
A finite probability exists that the particle can be
found outside the well even if E < U
The uncertainty principle allows the particle to be
outside the well as long as the apparent violation of
conservation of energy does not exist in any
measurable way
Finite Potential Well –
Region II
U=0
The allowed wave functions are sinusoidal
The boundary conditions no longer require
that ψ be zero at the ends of the well
The general solution will be
ψII(x) = F sin kx + G cos kx
where F and G are constants
Finite Potential Well –
Regions I and III
The Schrödinger equation for these
regions may be written as
d 2ψ 2m (U − E )
=
ψ
2
2
dx
h
The general solution of this equation
is
ψ = AeCx + Be −Cx
A and B are constants
Finite Potential Well –
Regions I and III, cont.
In region I, B = 0
In region III, A = 0
This is necessary to avoid an infinite value
for ψ for large negative values of x
This is necessary to avoid an infinite value
for ψ for large positive values of x
The solutions of the wave equation
become
Cx
−Cx
Ψ I = Ae for x < 0 and Ψ III = Be for x > L
Finite Potential Well –
Graphical Results for ψ
The wave functions for
various states are
shown
Outside the potential
well, classical physics
forbids the presence of
the particle
Quantum mechanics
shows the wave
function decays
exponentially to
approach zero
Finite Potential Well –
Graphical Results for ψ2
The probability
densities for the
lowest three
states are shown
The functions are
smooth at the
boundaries
Finite Potential Well –
Determining the Constants
The constants in the equations can be
determined by the boundary conditions and
the normalization condition
The boundary conditions are
dψI dψII
ψI = ψII and
at x = 0
=
dx
dx
dψII dψIII
ψII = ψIII and
=
at x = L
dx
dx
Usually a lengthy calculation…
Boundary Conditions
ψI(x)=Aecx
ψII(x)=F sinkx +
ψIII(x)=Be-cx
ψI(0)= ψII(0)
ψI’(0)= ψII’(0)
ψII(L)= ψIII(L)
ψII’(L)= ψIII’(L)
G coskx
A=G
cA=kF
FsinkL+GcoskL=Be-cx
k(FcoskL-GsinkL)=-cBe-cx
Transcendental Equation
k(F – G tankL)/(F tankL + G) = -c
with c/k = F/A ≡ α
(α – tankL)/(α tankL + 1) = -α
It can be simplified to tan kL = -2α/(α2 – 1)
again a=c/k; k2=2mE/ћ2, c2=2m(Uo-E)/ћ2
There is no analytical solution, need to
solve numerically. The above equation
is often referred to as a “transcendental
equation”.
Solutions
For Uo=6E∞ where E∞ =h2/8mL2
Total E≡bE∞, α2=(6-b)/b, kL=π√b
20
RHS
tan
15
-2α/(α2-1)
Only 3 solutions
possible
10
tankL
5
0
-5
0
1
2
3
4
5
-10
-15
-20
0.625
2.43
b
(E=bEinf)
5.09
6
E1=0.625E∞
E2=2.43E∞
E3=5.09E∞
Eigenfunctions and Eigenvalues
Application – Nanotechnology
Nanotechnology refers to the design and
application of devices having dimensions
ranging from 1 to 100 nm
Nanotechnology uses the idea of trapping
particles in potential wells
One area of nanotechnology of interest to
researchers is the quantum dot
A quantum dot is a small region that is grown in
a silicon crystal that acts as a potential well
Potential barriers and tunneling
A fun novel to learn about tunneling is Mr. Thompkins in
Wonderland by Gamov. In the book, after attending a lecture on
quantum theory, Mr. Thompkins arrives home and parks in his
garage. He awakens to find his car out in the driveway!
Tunneling
The potential energy
has a constant value U
in the region of width L
and zero in all other
regions
This a called a square
barrier
U is the called the
barrier height
Tunneling, cont.
Classically, the particle is reflected by the
barrier
Regions II and III would be forbidden
According to quantum mechanics, all
regions are accessible to the particle
The probability of the particle being in a
classically forbidden region is low, but not zero
According to the uncertainty principle, the
particle can be inside the barrier as long as the
time interval is short and consistent with the
principle
Tunneling, final
The curve in the diagram represents a full
solution to the Schrödinger equation
Movement of the particle to the far side of
the barrier is called tunneling or barrier
penetration
The probability of tunneling can be
described with a transmission coefficient,
T, and a reflection coefficient, R
Tunneling Coefficients
The transmission coefficient represents the
probability (ψ*ψ) that the particle penetrates to the
other side of the barrier
The reflection coefficient represents the probability
that the particle is reflected by the barrier
T+R=1
The particle must be either transmitted or reflected
T ≈ e-2CL and can be nonzero
Tunneling is observed and provides evidence of the
principles of quantum mechanics
Applications of Tunneling
Alpha decay
In order for the alpha particle to escape from
the nucleus, it must penetrate a barrier whose
energy is several times greater than the energy
of the nucleus-alpha particle system
Nuclear fusion
Protons can tunnel through the barrier caused
by their mutual electrostatic repulsion
Alpha Decay
Potential Barrier
Examples of tunnelling
Tunnelling occurs in many situations in physics and astronomy:
1. Nuclear fusion (in stars
and fusion reactors)
V
Strong
nuclear force
(attractive)
Coulomb
interaction
(repulsive)
Incident
particles
2. Alpha-decay
V
Internuclear
distance x
( Ze) 2
Barrier height ~
~ MeV
4πε 0 rnucleus
thermal energies (~keV)
Initial αparticle energy
Distance x of αparticle from
nucleus
More Applications of Tunneling –
Scanning Tunneling Microscope
An electrically conducting
probe with a very sharp
edge is brought near the
surface to be studied
The empty space between
the tip and the surface
represents the “barrier”
The tip and the surface are
two walls of the “potential
well”
Scanning Tunneling Microscope
The STM allows
highly detailed
images of surfaces
with resolutions
comparable to the
size of a single atom
At right is the
surface of graphite
“viewed” with the
STM
Scanning Tunneling
Microscope, final
The STM is very sensitive to the distance
from the tip to the surface
This is the thickness of the barrier
STM has one very serious limitation
Its operation is dependent on the electrical
conductivity of the sample and the tip
Most materials are not electrically conductive at
their surfaces
The atomic force microscope overcomes this
limitation
Real atoms in a 2-D well
Following our model,
surface Scientists
placed 48 Fe atoms
on a Cu surface
(this is done at
ultra-high vacuum
and the image is
made with
scanning-tunneling
electron
microscopy).
Tunneling Microscope
Simple Harmonic Oscillator
Reconsider black body radiation as vibrating
charges acting as simple harmonic oscillator
Important because many systems can be
approximated by SHO (small oscillations)
The potential energy is
U = ½ kx2 = ½ mω2x2
Its total energy is
E = K + U = ½ kA2 = ½ mω2A2
SHO – ground state
The Schrödinger equation for this problem is
h 2 d 2ψ 1
2 2
−
+
mω
x ψ = Eψ
2
2m dx
2
The ground state solution of this equation is
− ( mω 2 h ) x 2
B is a constant
ψ = Be
where c = mω/2ћ
ψ’ = -2cxBe−cx2
ψ’’ = -2Be-cx2 + 4c2x2Be-cx2 = -2cψ + 4c2x2ψ
So the Schrödinger equation becomes
-2cψ + 4c2x2ψ - 4c2x2ψ = −(2mE/ћ2)ψ
or E = cћ2/m = ½ ћω (ground state energy)
General Solution - SHO
The remaining solutions that describe the excited states all
include the exponential function
2
ψn(x) = Bn(x)e-Cx
The energy levels of the oscillator are quantized.
Solution of Bn(x) is related to the Hermite Polynomials
The energy for an arbitrary quantum number n is
En = (n + ½)ћω where n = 0, 1, 2,…
The exact wave function is
ψ n (u ) =
where
c
x
2
u=
1
−u 2 / 2
H n ( u )e
n 1/ 2
n!2 π
Hermite Differential Equation
y’’ – 2xy’ + 2ny = 0
Hn(x) = (-1)nex2dn/dxn(e-x2)
Hn+1(x) = 2xHn(x) – 2nHn-1(x)
H0(x)
H1(x)
H2(x)
H3(x)
H4(x)
H5(x)
=
=
=
=
=
=
1
2x
-2 + 4x2
-12x + 8x3
12 – 48x2 + 16x4
120x – 160x3 + 32x5
Energy Level Diagrams –
Simple Harmonic Oscillator
The separation between
adjacent levels are equal
and equal to ∆E = ћω
The energy levels are
equally spaced
The state n = 0
corresponds to the ground
state
The energy is Eo = ½ ћω
Agrees with Planck’s
original equations
Comparison of Newtonian and
Quantum Oscillators
Quantum mechanics in 3-D
•
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The atomic/molecular
world is a 3-D place.
Cartesian coordinates
work but are
computationally
cumbersome. The
system that works
best with spherical
coordinates.
But “2 becomes
complicated.
Question
The graph below represents a wave function
for a particle confined to . The value of the
normalization constant A may be
A
–2
a.
b.
c.
d.
e.
-1/4
-1/2
1/2
1/4
either -1/2 or +1/2
+2
x
Question
The graph below represents a wave function
for a particle confined to . The value of the
normalization constant A may be
A
–2
a.
b.
c.
d.
e.
-1/4
-1/2
1/2
1/4
either -1/2 or +1/2
+2
x
Question
The first five wave functions for a
particle in a box are shown. The
probability of finding the particle
near x = L/2 is
A. least for n = 1.
B. least for n = 2 and n = 4.
C. least for n = 5.
D. the same (and nonzero) for n = 1,
2, 3, 4, and 5.
E. zero for n = 1, 2, 3, 4, and 5.
Question
The first five wave functions for a
particle in a box are shown. The
probability of finding the particle
near x = L/2 is
A. least for n = 1.
B. least for n = 2 and n = 4.
C. least for n = 5.
D. the same (and nonzero) for n = 1,
2, 3, 4, and 5.
E. zero for n = 1, 2, 3, 4, and 5.
Question
The first five wave
functions for a particle in a
box are shown. Compared
to the n = 1 wave function,
the n = 5 wave function has
A. the same kinetic energy (KE).
B. 5 times more KE.
C. 25 times more KE.
D. 125 times more KE.
E. none of the above
Question
The first five wave
functions for a particle in a
box are shown. Compared
to the n = 1 wave function,
the n = 5 wave function has
A. the same kinetic energy (KE).
B. 5 times more KE.
C. 25 times more KE.
D. 125 times more KE.
E. none of the above
Question
The first three wave functions
for a finite square well are
shown. The probability of
finding the particle at x > L is
A. least for n = 1.
B. least for n = 2.
C. least for n = 3.
D. the same (and nonzero) for
n = 1, 2, and 3.
E. zero for n = 1, 2, and 3.
Question
The first three wave functions
for a finite square well are
shown. The probability of
finding the particle at x > L is
A. least for n = 1.
B. least for n = 2.
C. least for n = 3.
D. the same (and nonzero) for
n = 1, 2, and 3.
E. zero for n = 1, 2, and 3.
Question
A potential-energy function is
shown. If a quantum-mechanical
particle has energy E < U0, it is
impossible to find the particle in
the region
A. x < 0.
B. 0 < x < L.
C. x > L.
D. misleading question — the
particle can be found at any x
Question
A potential-energy function is
shown. If a quantum-mechanical
particle has energy E < U0, it is
impossible to find the particle in
the region
A. x < 0.
B. 0 < x < L.
C. x > L.
D. misleading question — the
particle can be found at any x
Question
The figure shows the first six energy
levels of a quantum-mechanical
harmonic oscillator. The
corresponding wave functions
A. are nonzero outside the region
allowed by Newtonian mechanics.
B. do not have a definite wavelength.
C. are all equal to zero at x = 0.
D. Both A. and B. are true.
E. All of A., B., and C. are true.
Question
The figure shows the first six energy
levels of a quantum-mechanical
harmonic oscillator. The
corresponding wave functions
A. are nonzero outside the region
allowed by Newtonian mechanics.
B. do not have a definite wavelength.
C. are all equal to zero at x = 0.
D. Both A. and B. are true. (why?)
E. All of A., B., and C. are true.
Question
θ
θ(0) = 5o
release from rest
θ(t) = ?
What does the
differential equation looks like?
1 kg
Example: the quantum well
Quantum well is a “sandwich” made of two different semiconductors in which the
energy of the electrons is different, and whose atomic spacings are so similar that
they can be grown together without an appreciable density of defects:
Material A
(e.g. AlGaAs)
Material B (e.g. GaAs)
Electron
potential energy
Position
Now used in many electronic devices (some
transistors, diodes, solid-state lasers)
Kroemer
Esaki