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Chapter 38
Quantum Mechanics
Units of Chapter 38
38-1 Quantum Mechanics – A New Theory
37-2 The Wave Function and Its Interpretation; the
Double-Slit Experiment
38-3 The Heisenberg Uncertainty Principle
38-4 Philosophic Implications; Probability versus
Determinism
38-5 The Schrödinger Equation in One Dimension –
Time-Independent Form
38-6 Time-Dependent Schrödinger Equation
Quantum Mechanics – A New Theory
In the early 1920s, it became increasingly
evident that a new, more comprehensive theory
was needed.
The new theory, called quantum mechanics,
has been extremely successful in unifying into
a single consistent theory
the wave-particle duality, black-body radiation,
atoms, molecules, and many other phenomena.
It is widely accepted as being the fundamental
theory underlying all physical processes.
The Wave-Particle Duality
⎛ p ⎞
⎜
⎟
⎝ E c ⎠
h
=
2π

⎛ k ⎞
⎜
⎟
⎝ ω c ⎠
If waves can behave like particle, then
particles can behave like waves
h
λ=
p
De Broglie wavelength
The Wave Nature of Matter
The properties of waves, such as interference
and diffraction, are significant only when the
size of objects or slits is not much larger than
the wavelength.
If the mass is really small, the wavelength can
be large enough to be measured.
h
λ=
p
The Wave Nature of Matter
Photon diffraction
Electron
diffraction
38.2 The Wave Function and Its
Interpretation; the Double-Slit Experiment
What is oscillating in a matter wave?
It is the probability of finding the particle
that waves.
A matter wave is described by the wave function,
Ψ. The square of the wave function |Ψ| 2
(probability distribution) at any point is
proportional to the number of particles expected
to be found there. For a single particle, the wave
function is the probability of finding the particle
at that point.
38.2 The Wave Function and Its
Interpretation; the Double-Slit
Experiment
The interference pattern
is observed after many
electrons have gone
through the slits. If we
send the electrons
through one at a time, we
cannot predict the path
any single electron will
take, but we can predict
the overall distribution.
Double-Slit Experiment
with Electrons
Figure 38.4
Double-Slit Experiment
with Electrons
|Ψ| represents the matter
wave amplitude and
|Ψ|2 represents the probability
of finding a given electron at
a given point.
38.3 The Heisenberg Uncertainty
Principle
Quantum mechanics tells us there are limits to
measurement – not because of the limits of our
instruments, but inherently.
This is due to wave-particle duality, and to
interaction between the observing equipment
and the object being observed.
38.3 The Heisenberg Uncertainty
Principle
Imagine trying to see an
electron with a powerful
microscope. At least one
photon must scatter off the
electron and enter the
microscope, but in doing
so it will transfer some of
its momentum to the
electron.
38.3 The Heisenberg Uncertainty
Principle
The uncertainty in the momentum of the
electron is taken to be the momentum of the
photon – it could transfer anywhere from none
to all of its momentum.
Δpx ≈ h λ
In addition, the position can only be measured
to about one wavelength of the photon.
Δx ≈ λ
38.3 The Heisenberg Uncertainty
Principle
The combination of uncertainties gives:
h
(Δx) (Δpx ) ≥
2π
which is called the
Heisenberg uncertainty principle.
It tells us that the position (x) and
momentum (p) cannot be measured with
infinite precision at the same time.
38.3 The Heisenberg Uncertainty
Principle
The uncertainty principle applies also to time
and energy:
h
(ΔE) (Δt) ≥
2π
This says that if an energy state only lasts for a
limited time, its energy will be uncertain. It also
says that conservation of energy can be
violated if the interaction time is short enough.
38.3 The Heisenberg Uncertainty
Principle
The uncertainty principle applies also to angular
variables
h
(ΔLz ) (Δφ) ≥
2π
38.3 The Heisenberg Uncertainty
Principle
h
(Δx) (Δpx ) ≥
2π
h
(ΔE) (Δt) ≥
2π
h
(ΔLz ) (Δφ) ≥
2π
The uncertainty principle states a
fundamental property of quantum systems,
and is not a statement about the
observational success of current technology.
38.4 Philosophic Implications;
Probability versus Determinism
The world of Newtonian mechanics is a
deterministic one. If you know the forces on an
object and its initial velocity, you can predict
where it will go.
Quantum mechanics is very different – you can
predict what ensembles of electrons will do,
but have no idea what any individual one will
do.
38-6 The Time-Dependent
Schrödinger Equation
What is the equation of motion of Ψ ?
∂2 D 1 ∂2 D
One-Dimensional Wave Equation
− 2 2 =0
2
∂x
v ∂t
Harmonic Wave
D(x,t) = A sin(kx − ω t) → A ei(kx − ω t )
1D EM Wave Equation in Vacuum:
∂2 E 1 ∂2 E
− 2 2 =0
2
∂x
c ∂t
What is the equation of motion of Ψ ?
38-6 The Time-Dependent
Schrödinger Equation
What is the equation of motion of Ψ ?
⎡  2 ∂2
⎤
∂
⎢ − 2m ∂x 2 + V (x) ⎥ Ψ(x,t) = i ∂t Ψ(x,t)
⎣
⎦
so-called, Time-Dependent Schrödinger Equation
This equation is satisfied by a harmonic wave function in
the special case of a free particle (no net force acts)
V (x) = V0
38-6 The Time-Dependent
Schrödinger Equation
⎡  2 ∂2
⎤
∂
⎢ − 2m ∂x 2 + V0 ⎥ Ψ(x,t) = i ∂t Ψ(x,t)
⎣
⎦
Ψ = A ei(kx − ω t )
2 k 2
+ V0 = ω
2m
→
The physical significance of the wave
function Ψ is associated with the
probability density
p2
+ V0 = E
2m
ΨΨ= Ψ
*
2
The probability of finding a particle
2
P(x,t) dx = Ψ dx
within a position range dx is
38-6 The Time-Dependent
Schrödinger Equation
Since the solution to the Schrödinger equation is
supposed to represent a single particle, the total
probability of finding that particle anywhere in
space should equal 1.
Normalization condition:
∫
all space
P(x, t) dV = 1
∞
one
dimension

∫
−∞
2
Ψ dx = 1
One-Dimensional Wave Equation
∂
1 ∂
D(x,t) = 2 2 D(x,t)
2
∂x
v ∂t
2
2
One-Dimensional Schrödinger Equation
⎡  2 ∂2
⎤
∂
⎢ − 2m ∂x 2 + V (x) ⎥ Ψ(x,t) = i ∂t Ψ(x,t)
⎣
⎦