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LECTURE 3 – BASIC QUANTUM THEORY
Matter waves and the wave function
In 1924 De Broglie proposed that all matter has a wavelength and exhibits wave like
behavior. He proposed that the wavelength of a particle of momentum p is
λ = 2π
!
p
where P is the momentum of the particle
The incentive for this expression was the corresponding one for photons proposed by
Einstein.
He described an electron as a standing wave around the circumference of an atomic orbit
such that,
nλ = 2πr
This was combined with Niels Bohr’s postulate that angular momentum was quantized
(which we will later see implies a limited number of energies),
to give a wavelength
mvr = n!
λ=
Lecture 3: Basic Quantum Theory
2π!
p
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e.g.,
Calculate the de Broglie wavelength of an electron which has a kinetic energy of 4 eV
E = (1/2)mv 2 = (1/ (2m )) p 2
λ=
h
h
=
= 6.13 A
p
2mE
The wavelength has the dimensions of interatomic spacing.
What’s the de Broglie wavelength of a tennis ball having a mass of 50 g and traveling
with a velocity of 200 km/h?
Similar calculations show that
λ = 2.38 × 10
− 24 A
If electrons are just particles, they should behave as such:
Behavior of a stream of particles and a wave impinging on a screen with a small opening.
The wave is said to be diffracted, diffraction is observed when a wave is distorted by an
obstacle which has dimensions comparable to the wavelength of the wave.
Lecture 3: Basic Quantum Theory
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The wave like behavior of electrons was experimentally confirmed by the Bragg
diffraction of electrons by crystals. This phenomenon is analogous to crystal diffraction
of an EM wave (X-rays).
Bragg picture of X- ray Diffraction
Reflection is in phase if nλ = 2dsinθ (Bragg condition), and we get constructive
interference and a diffraction pattern is obtained.
It was observed that the same diffraction pattern was produced by X-rays of wavelength
λ = h/p and that produced by electrons with the same de Broglie wavelength.
nλ = 2dsinθ = nh/p
This is a result of quantum theory, and holds for photons for relativistic and nonrelativistic particles (with a velocity small compared with that of light). Other particles
include neutral atoms, protons, positive ions, etc.
Lecture 3: Basic Quantum Theory
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Back to Quantum Mechanics
Unified scheme of Mechanics based upon notions of Planck’s quantum theory (E = hν)
and utilizing de Broglie’s ideas of the wave like nature of matter. It is a revision of the
“laws of mechanics” design to extend the subject into the realm of atomic and nuclear
phenomena.
Classically an EM wave is interpreted as changing electrical and magnetic fields. If a
particle has a wavelength, what is the field? What is meant by a wave to be associated
with a particle?
Born showed that the wave amplitude is related to the probability of locating the particle
in a given region of space. More specifically, in quantum mechanical problems, we
attempt to find a quantity Ψ called the wave function. While Ψ itself has no direct
physical meaning, it is defined in such a way that the probability of finding the particle in
the region of space between x and x + dx and y and y + dy, and z and z + dz is given by
ψ * ψ dx dy dz
The interpretation of the wavelength of an electron is through the function Ψ. Since the
probability that the particle will be found somewhere in space is unity, we must require
that the wave function be normalized such
∫ψ ψ dx dy dz =1
*
the integral being taken over all space.
This makes it clear that in quantum mechanics probability statements are often obtained,
whereas in classical mechanics the location of a particle can be determined exactly.
The Schrodinger Equation
Solution to the calculation and interpretation of Ψ provided by Schrodinger in 1925:
Motion of electron with energy E described by a “wavefunction”:
The properties of the wave function can be expressed in the form of postulates:
1.- Associated with the particle there is a complex wave function Ψ(x,y,z,t), where x,y,z
are space coordinates and t is the time.
*
*
Ψ (r , t ) = Ψ (r )e
Lecture 3: Basic Quantum Theory
−
jωt
(1)
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&
where, Ψ (r ) - position dependent part
e
−
jωt
- time dependent part
2.- The classical expression for the total energy (Et), given by
p2
+ v( X , Y , Z )
Et =
2m
where p is the momentum of the particle, m its mass and V(x,y,z) its potential energy,
may be converted into a wave equation by associating certain operators. Let’s define the
momentum operator
!
p= ∇
i
such that when substituted in the expression for Et, we get
!2 2
1
2
(− i!∇ ) + v( X , Y , Z ) = − ∇ + v( X , Y , Z ) .
Et =
2m
2m
When we “operate” with this expression on Ψ, we obtain,
− !2 2
&
∇ Ψ + V(r ) = E
2m
This is the Time Independent Schrodinger Equation and is obeyed by Ψ(x,y,z).
∂ 2Ψ ∂ 2Ψ ∂ 2Ψ
∇ Ψ= 2 + 2 + 2
∂x
∂y
∂z
2
2
where, ∇ - Laplacian operator defined before.
What does Ψ mean? What is the interpretation of the wave function?
Lecture 3: Basic Quantum Theory
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&
&
Ψ (r , t ) ⋅ Ψ * (r , t ) = probability density (per unit volume)
&
of finding electron at position r at time t
*
where, Ψ * (r , t ) - complex conjugate.
Schrodinger equation is a guess; it can’t be derived but it has been confirmed
experimentally.
Solution to Schrodinger equation for special problems
Free Electrons
&
" Free" ⇒ V(r ) = 0 everywhere
 !2  2
∇ Ψ = EΨ
 −
 2m 
∴
(5)
 !2  d 2Ψ
 2 = EΨ
consider 1 - dim case :  −
 2m  dx
Solution : ψ ( x ) = Ce jkx where C is a constant
(6)
(7)
d2Ψ
= −k 2 Ψ
2
dx
∴ need
!2k2
=E
2m
classical physics :
p2
=E
2m
or
k=
2mE
!
(8)
∴ identify !k = electron momentum = p
Ψ ( x, t ) = Ce jkx e jωt = Ce j ( kx− wt )
(9)
(10)
This is the equation of a travelling wave. It represents a free particle of momentum and
energy given by Equation 9.
Lecture 3: Basic Quantum Theory
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Phase velocity defined as: k∆x = ω∆t
Consider a harmonic plane wave, the phase velocity is the speed at which the profile
moves:
ψ
t=0
time ∆t later
∆X
x
λ
The phase velocity can also be expressed as:
Phase velocity =
∆x ω E
E
!
= = ⋅
=
∆t k ! 2mE
2m
More on Plane waves:
Lecture 3: Basic Quantum Theory
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Harmonic functions should repeat themselves in space after a displacement of λ in the
direction of propagation (K). Then,
ψ (r ) = ψ (r +
λK
)
k
where k is the magnitude of K and K/k is a unit vector parallel to it, then
AeiK ⋅r = AeiK ⋅( r + λK / k ) = AeiK ⋅ r eiλk
for this to be true
eiλk = 1 = ei 2π
and finally
λ=
2π
k
We can generalize to 3 dim:
 !2  2
∇ Ψ = EΨ
 −
2
m


with
& &
Ψ = Ce jk⋅ r
( 11)
&
where, k - wave vector
& 2π
k =
λ
* *
&
Ψ (r , t ) = Ce j (k ⋅r −ωt )
(12)
- Travelling plane wave solution
- Wave moves in direction of k
- Planes of constant phase perpendicular to k
- Wavelength given by,
2π
λ=
k
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