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Wave mechanics and its applications
Contents
1 Schrödinger equation
1.1 Mathematical representation . . . . . . . . . . . . . . . . . . .
1
2
2 Quantum mechanical operators
3
3 Particle in a potential well
3.1 Numerical example . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Extension to 3D . . . . . . . . . . . . . . . . . . . . . . . . . .
5
7
8
4 Quantum tunneling
9
4.1 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . 13
1
Schrödinger equation
Wave mechanics is an approach to solving quantum mechanical systems, to
obtain properties of the system e.g. energy and momentum. There are other
approaches, matrix mechanics being an earlier one that was replaced by wave
mechanics due to its simplicity. In wave mechanics the quantum particle,
usually the electron (for most condensed matter studies) is represented by a
wavefunction. The concept of what the wave and hence the wavefunction
represents will be dealt with later.
The governing equation in wave mechanics is the Schrödinger equation.
This is primarily an energy equation written as
KE + PE = Total energy
1
(1)
MM3010: Physics of materials
In classical mechanics, for a particle with mass m and velocity v, equation 1
can be written as
1 2
mv + V = E
2
(2)
p2
+ V = E
2m
where p is the momentum. In quantum mechanics the particle is replaced by
a wavefunction and this representation is used in the energy equation.
1.1
Mathematical representation
To obtain the mathematical representation of the Schrödinger equation consider a one dimensional electron wave (this can be any quantum particle but
we will stick to electrons) that is written as
ψ = A exp i(kx − ωt)
(3)
here A is a normalization constant, k is called the electron wavevector and
ω is the angular momentum. The kinetic energy term can be related to the
wavefunction through the wavevector term. This comes from the de Broglie
relation given by
h
2π
=
k
p
(4)
hk
= ~k
p =
2π
To obtain k in terms of the wavefunction ψ consider the second derivative of
ψ with respect to x
∂ 2ψ
= −k 2 ψ
(5)
∂x2
Using equation 4 the kinetic energy can then be written in terms of the
wavefunction as
λ =
p2
~2 k 2
=
2m
2m
2
~ ∂ 2ψ
KE = −
2mψ ∂x2
KE =
(6)
The potential energy can be written in a general form as V (x) (for the onedimensional case). The total energy, E, can be written as ~ ω. Again it is
possible to write ω in terms of ψ by
∂ψ
= −iωψ
∂t
2
(7)
MM3010: Physics of materials
So the total energy term now becomes
E = ~ω = −
~ ∂ψ
iψ ∂t
(8)
Equations 6 and 8 can then be substituted into 1 and rearranged to obtain
the one-dimensional time dependent Schrödinger equation
−
~2 ∂ 2 ψ
∂ψ
+ V (x) ψ = i~
2
2m ∂x
∂t
(9)
Most system that we will consider are steady state (so time independent) so
that equation 9 becomes
−
~2 d2 ψ
+ V (x) ψ = E ψ
2m dx2
(10)
where E is the total energy, that is a constant. This is the one-dimensional
time independent Schrödinger equation and can be extended to three dimensions and written as
−
~2 2
∇ ψ+ V ψ = E ψ
2m
(11)
∇2 is called the Laplacian operator and depends on the coordinate system.
For the Cartesian coordinate system
∇2 =
∂2
∂2
∂2
+
+
∂x2
∂y 2
∂z 2
(12)
V the potential energy and is generally time independent but dependent on
the position e.g. V (x, y, z). The Schrödinger equation, either equation 10 or
11, can be solved for the specific system to obtain the solution of the electron
wavefunction.
2
Quantum mechanical operators
It is possible to define measurable quantities like position, momentum (velocity), and energy for a classical particle. The same can be done for an electron
(or any other quantum entity) defined by a wavefunction ψ. But first we need
to understand the meaning of the wavefunction or an electron wave. ψ does
not represent a physical wave (like water waves or sound waves) but rather
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MM3010: Physics of materials
Table 1: Some commonly used quantum mechanical operators
Position
Linear momentum (similarly for py and pz )
Kinetic energy
Potential energy
Total energy
Hamiltonian
Symbol QM operator
x
x
~ ∂
px
i ∂x
~2 2
KE
−
∇ (3D)
2m
V
V
∂
E
i~
∂t
~2 2
H
∇ + V
2m
represents a probability wave. More generally |ψ(x)|2 represents the probability density i.e. probability per unit length (since this is a one-dimensional
system) of an electron at a certain position x. The probability can be written
as
Z
Z
2
ψ ∗ (x) ψ dx
(13)
|ψ(x)| dx =
Probability =
ROI
ROI
where ROI represents the region of interest and determines the limits of the
integral. ψ ∗ is the complex conjugate of the electron wavefunction ψ. For
a three dimensional wavefunction dx is replaced by a volume element dV . A
wavefunction is said to be normalized if the total probability of finding it
over all space is one.
Just like calculating the probability it is possible to calculate other properties
by using what are called quantum mechanical (QM) operators. If A is
a QM operator ‘acting’ on a wavefunction ψ then if it is possible to write
Aψ = λψ
(14)
where ψ is called the Eigen function of the operator A and λ is then
called the Eigen value of A. Some of the commonly used QM operators are
listed in table 1. Thus, if ψ(x) is a one-dimensional electron wavefunction to
calculate it momentum use the operator from table 1.
~ ∂ψ(x)
= λψ(x)
(15)
i ∂x
Here λ represents the Eigen value of the momentum operator.
QM operators can also be used to calculate average or expectation values.
Thus, if A is a QM operator the average value of that property is given by
R ∗
ψ A ψ dx
<A>= R ∗
(16)
ψ ψ dx
px =
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MM3010: Physics of materials
The integration is performed over all space. The denominator is the normalization factor since it represents the probability of finding the electron over
all space (should be one). Thus, to find the average position of the electron
replace A by the corresponding QM operator from table 1 and equation 16
becomes
R ∗
ψ x ψ dx
(17)
<x>= R ∗
ψ ψ dx
Similarly the average kinetic energy (one dimensional) can be written as
2
< KE > = −
~
2m
R
d2 ψ
ψ∗
dx
2
R dx
ψ ∗ ψ dx
(18)
The electron wavefunction, ψ, is a solution to the Schrödinger equation. For
this to represent a physical system, ψ must have certain properties. It must
be single-valued, continuous, and finite throughout the system. To gain a
better understanding of Schrödinger equation and QM operators we can look
at its application for a few simple systems.
3
Particle in a potential well
The simplest system to apply the one dimensional Schrödinger equation is a
particle (electron) in a constant potential region. For simplicity this constant
potential can be set as zero. To obtain physical solutions it is easier to give
a finite dimension to this region of zero potential i.e. potential well. Let a
be the length of the well, depicted in figure 1.
To make sure that the particle does not ‘escape’ the well the walls can
be made of infinite potential. Since the electron cannot escape the well, we
only need to write the Schrödinger equation for the electron in the region
0 < x < a. This can be written, by modifying equation 10 with V = 0.
2mE
d2 ψ
+
ψ = 0
2
dx
~2
(19)
This is a standard wave equation and the general solution is written as
ψ = A exp(ikx) + B exp(−ikx)
where A, B, and k are constants. k is given by
r
2mE
k =
~2
5
(20)
(21)
MM3010: Physics of materials
Figure 1: Schematic of the one dimensional potential well. Electrons are
trapped inside a potential well of width a. The walls have infinite height
while within the well the potential is zero
k is called the wavevector.To calculate the other constants we need to make
use of boundary conditions. Given that the walls have an infinite potential,
the particle will not be able to ‘escape’ the well. Hence the wavefunction
must go to zero at the boundaries i.e. ψ(0) = ψ(a) = 0. Substituting this
boundary condition in equation 20 gives
A + B = 0 ⇒ A = −B
A exp(ika) + B exp(−ika) = 0 ⇒ 2iA sin(ka) = 0
(22)
The second boundary condition in equation 22 gives the relation between k
and a i.e. ka = nπ with n being a non-zero positive integer. Substituting
in equation 20 gives the general solution
ψ(x) = 2iA sin(kx) ⇒ ψ(x) = 2iA sin(
nπx
)
a
(23)
To evaluate A we can use the fact the wavefunction must be normalized i.e.
the probability of finding the electron somewhere within the well should be
1. This can be expressed mathematically as
Z a
ψ ∗ (x) ψ(x) dx = 1
(24)
0
Substituting the wavefunction, from equation 23, into 24 the constant A can
be calculated to be
r
1
A =
(25)
2a
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MM3010: Physics of materials
Table 2: Electron energies for the first few levels of a potential well of width
2 nm.
n Energy (eV)
1
0.09
2
0.36
3
0.81
4
1.44
5
2.25
Thus, the final normalized wavefunction for an electron (particle) in a box is
given by
r
2
nπx
ψ(x) = i
sin(
)
(26)
a
a
This wavefunction is said to be quantized since it can only take discrete values
for n (positive non-zero integers). n is thus called a quantum number for an
electron in the one-dimensional potential well. To calculate the total energy
we can either use the Hamiltonian operator, used in table 1, or substitute the
wavefunction, equation 26, in the Schrödinger equation, equation 19. This
gives En to be
n2 h2
(27)
En =
8ma2
Thus, the energy of the electron in a potential well is also quantized.
3.1
Numerical example
Consider an electron ‘trapped’ in a potential well of width a = 2 nm. Using
standard values for mass of the electron me and h the lowest energy of the
electron is given by equation 27 to be 1.51 × 10−20 J or 0.09 eV or 90 meV .
This corresponds to n = 1 in equation 27. The energies of electrons in the
first few energy levels are given by table 2. To excite an electron from n = 1
to n = 2 The minimum ebergy required is given by the difference between
their energies, table 2. Also, when an electron ‘drops down’ from a higher
level to a lower level the difference in energies is given out as light (EM
radiation). The magnitude of the energy levels are inversely proportional to
the square of the width, see equation 27. Thus, as a decreases, En , increases.
The wavefunction of the electron can be obtained by using 26 by substituting
different values for n. Because of the presence of i this wavefunction is an
imaginary function. This can be converted into a real quantity, i.e. probability
density by multiplying with its complex conjugate. This gives the probability
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MM3010: Physics of materials
Figure 2: Plot of the (a) wavefunction and (b) probability density for the
first 4 levels for a particle in a 1D well. The red horizontal line represents
the origin. Plot generated using MATLAB.
of finding the electron, per unit length, for different values of n. Using
equation 26 this gives
|ψ(x)|2 = ψ(x)∗ ψ(x) =
nπx
2
sin2 (
)
a
a
(28)
Both the wavefunction and the probability density are plotted for values of
n from 1-4 in figure 2 The horizontal line for each plot corresponds to the
x-axis. For n = 1 the wavefunction goes to zero only at the edges. For n = 2
ψ(x) goes to zero at x = 0.5a i.e. the center (apart from the ends). This
point is called a node. For n = 3 there are 2 nodes and in general there are
(n − 1) nodes for a given quantum number, n.
3.2
Extension to 3D
The one dimensional particle in a well model can be extended to a particle in
a three dimensional well. The boundary conditions remain the same, except
that the three dimensional Schrödinger equation is now used, equation 11.
For Cartesian coordinates the Laplacian operator is given by equation 12.
Since the potential is zero, it is possible to separate Schrödinger equation
into three separate equations, one for each coordinate. Since each of this is
a one dimensional equation, the solution is similar to equation 26. Thus the
three dimensional wavefunction, ψ(x, y, z), can be written as
ψ(x, y, z) = A sin(
ny π
nz π
nx π
) sin(
) sin(
)
a
b
c
8
(29)
MM3010: Physics of materials
where A is a constant that can be obtained by normalization and a, b, and c
are the dimensions of the box along the three directions. Similar to equation
27 it is possible to write the total energy of the electron as
E(nx ,ny ,nz )
h2
=
8m
n2y
n2x
n2z
+
+
a2
b2
c2
(30)
The energy is quantized and depends on a set of three positive non-zero
integers (nx , ny , nz ). These are called the quantum numbers for the electron
in the three dimensional box. For a particle in a three dimensional box it
is possible to have different sets of quantum numbers with the same energy.
For example when a = b = c equation 30 reduces to
E(nx ,ny ,nz )
h2
[n2x + n2y + n2z ]
=
2
8ma
(31)
In this case energy levels with quantum numbers (1, 1, 2), (1, 2, 1), and (2, 1, 1)
all have the same energy. These states are called degenerate states.
The particle in a box model, while particularly simple, can be used to explain
the optical properties of spatially confined systems. These are called quantum dots (three dimensional confinement), quantum wires (two dimensional
confinement), or thin films (one-dimensional confinement). Electrons in such
systems can be considered as a particle ‘trapped’ in a uniform potential well.
The energy levels of the electron is then quantized, and the actual values depend on the dimensions of the well. Transitions between these energy levels
can cause light emission with the color depending on the size of well.
4
Quantum tunneling
A particular application of Schröndinger equation is in solving the problem of
tunneling, which has no classical analogue. A one dimensional representation
of the system is shown in figure 3.
An electron with energy E is incident, from the left, on a potential barrier
of height V0 and width a (E < V0 ). Using the laws of classical mechanics
there is zero probability for the electron to be present on the other side
of the barrier. But in quantum mechanics, using the delocalized electron
wavefunction it is possible to show that there is a finite probability for
the electron to ‘tunnel’ through the barrier. This probability depends on the
energy, E, barrier height, V0 and width, a.
Consider the one dimensional system shown in figure. There are 3 regions,
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MM3010: Physics of materials
Figure 3: Schematic of the one dimensional quantum tunneling problem.
Electrons of energy E are incident on a barrier of height V0 , with E < V0 .
marked I, II, and III. It is possible to write the Schrödinger equation for
each of these three regions, by modifying equation 10.
−
−
~2 dψI
= E ψI ⇒ Region I
2m dx
~2 dψII
+ V0 ψII = E ψII ⇒ Region II
2m dx
~2 dψIII
= E ψIII ⇒ Region III
−
2m dx
(32)
Here ψI , ψII , and ψIII are the electron wavefunctions in the three regions
respectively. It is possible to write a general solution, similar to equation 20,
for each of the regions.
r
2mE
ψI = A1 exp(ikI x) + B1 exp(−ikI x) ; kI =
~2
r
2m(E − V0 ) (33)
ψII = A2 exp(ikII x) + B2 exp(−ikII x) ; kII =
~2
r
2mE
ψIII = A3 exp(ikIII x) + B3 exp(−ikIII x) ; kIII =
= kI
~2
The pre-exponential terms, A1 to B3 are all constants. Consider ψI in equation 33. The first term represents a wave traveling from left to right (incident
wave) while the second term is a wave traveling from right to left (reflected
wave). Consider psiII , since E < V0 kII is imaginary. Hence ψII can be
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MM3010: Physics of materials
written as
r
ψII = A2 exp(−αx) + B2 exp(αx) ; α =
2m(V0 − E)
~2
(34)
Consider ψIII , once again the first term represents a wave traveling to the
right (incident wave) and the second term is a wave traveling to the left
(reflected wave). But there is no barrier to reflect the wave to the left, since
region III extends indefinitely. Hence the constant B3 = 0. To find the other
constants we can use the fact that the wavefunction must be continuous
must be continuous at the
and continuously differentiable i.e. ψ and dψ
dx
boundaries. Applying this to the solutions in equations 33 and 34 gives a set
of equations. At x = 0
ψI = ψII ; A1 + B1 = A2 + B2
dψII
dψI
=
; ikI (A1 − B1 ) = −α(A2 − B2 )
dx
dx
(35)
Similar at x = a
ψII = ψIII ; A2 exp(−αa) + B2 exp(αa) = A3 exp(ikIII a)
dψII
dψIII
=
; −α(A2 exp(−αa) − B2 exp(αa)) = iA3 kIII exp(ikIII a)
dx
dx
(36)
Consider the two parts of equation 36. It is possible to rewrite them so that
A2 exp(−αa) + B2 exp(αa) = A3 exp(ika)
ik
A2 exp(−αa) − B2 exp(αa) = − A3 exp(ika)
α
(37)
where k = kI = KIII and α is defined in equation 34. By adding and
subtracting the two terms iin equation 37 we can obtain A2 and B2 in terms
of A3 . This can be written as
A3
ik
A2 =
exp(ika) 1 −
exp(αa)
2
α
(38)
A3
ik
B2 =
exp(ika) 1 +
exp(−αa)
2
α
Similarly the two terms in equation 35 can be rewritten as
A1 + B1 = A2 + B2
α
A1 − B1 = − (A2 − B2 )
ik
11
(39)
MM3010: Physics of materials
Adding the two terms in equation 39 it is possible to write A1 in terms of A2
and B2 as
α
B2
α
A2
1 −
+
1 +
(40)
A1 =
2
ik
2
ik
Now substituting for A2 and B2 in terms of A3 using equation 38 the ratio
of A3 and A1 can be obtained.
A3
−2kα exp(−ikα)
=
A1
2kα cosh(αa) − i(α2 − k 2 ) sinh(αa)
where:
exp(αa) − exp(−αa)
sinh(αa) =
2
exp(αa) + exp(−αa)
cosh(αa) =
2
(41)
The ratio of the pre-exponential terms A3 and A1 is important because it
can be used to define the transmission coefficient (T ) of a barrier. This
defines the probability of an electron tunneling from region I to III. Another
definition of T is that if there is a flux of electrons incident on the potential
barrier in region I (current) then T determines the flux (current) measured
in region III, due to tunneling. The transmission coefficient is given by
T =
vIII |ψIII |2
vI |ψI (incident)|2
(42)
where vI and vIII are the velocities of the electrons in the two regions. Velocity depends on the value of k (due to the relation with energy) and since
the k values are same for regions I and III the velocities are the same. Thus,
T can be written as the ratio of the pre-exponential terms
T =
|A3 |2
|A1 |2
(43)
Using equation 41, it is possible to calculate this ratio and after simplification
the final term is
T =
1
V02
;
D
=
4E(V0 − E)
1 + D sinh2 (αa)
(44)
D is a constant that depends on the energies of the electron and the barrier
and α is defined earlier in equation 34. For a wide or tall barrier i.e. when
αa 1 equation 44 reduces to
T = T0 exp(−2αa) ; T0 =
12
16E(V0 − E)
V02
(45)
MM3010: Physics of materials
In equation 45 the exponential term dominates so that it can be re-written
as
T ≈ : exp(−2αa)
r
2m(V0 − E)
α =
~2
(46)
Typically, the barrier potential is taken to be a constant i.e. independent
of x. If the potential is a function of x, V (x), then it is possible to divide
into a large number of barriers, each with a constant potential. The overall
transmission coefficient is then a product of all the individual coefficients.
This is called the Gamow factor and for a wide or tall barrier with nonuniform potential, equation 46 can be written as
2
T ≈ exp −
~
Z
x2
p
dx 2m(V (x) − E)
(47)
x1
where x1 and x2 are the limits of the barrier region where V (x) > E and
V (x) is the potential function. For a constant potential equation 47 reduces
to equation 46.
4.1
Numerical example
Consider a potential barrier of width 2 nm. Let V0 = 10 eV and take
E − = 5 eV . These are typical values for electrons in metals. Using
equation 45 it is possible to calculate the transmission coefficient. For this
the value of α is first calculated using equation 34, and is 1.144 × 1010 m−1 .
Using α, T is then calculated to be 5.31 × 10−20 . Thus for every 2 × 1019
electrons incident on the barrier, 1 electron can tunnel through the barrier!
If the width is reduced to 1 nm, then T becomes 4.63 × 10−10 , i.e. an order of
ten increase in the current. To put this in perspective 1 nm is approximately
3 atoms wide while 2 nm is approximately 6 atoms wide (typical d-spacing
is 0.3 nm). The dependence of the transmission coefficient on barrier width
is shown in figure 4. This exponential dependence of tunneling on the width
(everything else remaining constant) is the principle behind the operation of
the scanning tunneling microscope (STM).
References
The following source materials were used for preparing this handout
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MM3010: Physics of materials
Figure 4: Plot of the transmission coefficient vs. barrier width for constant
E and V0 . (a) Normal plot (b) Semi log plot showing the exponential dependence. The transmission coefficient changes by over 17 orders of magnitude
when the width is changed from 0.3 nm to 2 nm! Plot generated using
MATLAB.
1. Principles of modern physics by Robert B. Leighton, McGraw-Hill
Book company, 1959.
2. Introduction to quantum mechanics by Linus Pauling and E. Bright
Wilson, CBS Publishers, 2008.
3. Quantum mechanics: theory and problems by S.L. Kakani and H.M.
Chandalia, Sultan Chand & Sons, 2004.
4. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles
by R. Eisberg and R. Resnick, Wiley Student edition, 2011
The PDF was prepared in LATEX, using Texmaker and MiKTeX compiler.
The plots were generated in MATLAB. Thanks to Prof. John H. Weaver,
University of Illinois at Urbana-Champaign, for comments and suggestions.
14