ENGPHYS 2QM3
INTRODUCTION TO
QUANTUM MECHANICS
Lecture 12
Subject Overview
Particle Properties of Particles
Wave Properties of Waves
1. Particle Properties of Light
2. Wave Properties of Particles
3. Quantum Mechanics
a) Schrödinger Equation
b) Potential wells and tunneling
4. Atomic Stability and Structure
5. Statistical Mechanics
6. Condensed Matter
What is ‘waving’ in the wavelike
behaviour of particles?
Historical Footnote
• Swiss physicist Felix Bloch recounted the story of how wave mechanics
came to be: One day, Nobel laureate Peter Debye said, "Schrödinger,
you are not working right now on very important problems anyway. Why
don't you tell us some time about that thesis of de Broglie, which seems
to have attracted some attention."
• And so Schrödinger did. He gave a talk about how French physicist
Louis de Broglie postulated that matter also has wave properties, but
Debye dismissed the talk as "childish," pointing out that "to deal
properly with waves, one had to have a wave equation."
• Schrödinger thought about it and set to work on wave mechanics.
• By the next talk, Schrödinger said, "My colleague Debye suggested that
one should have a wave equation; well, I have found one!"
• Years later, Bloch approached Debye and asked him about the
encounter. Debye claimed that he had forgotten, but Bloch thought that
he was regretful that he goaded Schrödinger into working out the
formula rather than doing it himself. Regardless, Debye turned to Bloch
and said, "Well, wasn't I right?"
Schrödinger's Equation
• One of the most unsatisfying aspects of any
undergraduate quantum mechanics course is
perhaps the introduction of the Schrödinger equation.
• After several lectures motivating the need for
quantum mechanics by illustrating the new
observations at the turn of the twentieth century,
usually the lecture begins with: “Here is the
Schrödinger equation.”
• Sometimes, similarities to the classical Hamiltonian
are pointed out, but no effort is made to derive the
Schrodinger equation in a physically meaningful way.
• This shortcoming is not remedied in the standard
quantum mechanics textbooks either.
Schrödinger's Equation
Here is the Schrödinger Equation:
iħ∂Ψ/∂t = –(ħ2/2m)∂2Ψ/∂x2 + VΨ
Linear 2nd order PDE
Written as 1-d; can be 2- or 3-d
Ψ is a function of x and t, i.e. Ψ(x,t)
Ψ(x,t) is a scalar and is a complex number
in general
Schrödinger's Equation
Schrödinger's equation cannot be derived
from other basic principles of physics; it is
a basic principle in itself.
i.e. we assert it
we test it (vs. experiment)
we begin to trust it
we look for flaws in it (via experiment)
we don’t find any
we begin to accept it.
Related Quote
"At first, people refuse to believe that a
strange new thing can be done, and then
they begin to hope it can be done, then
they see it can be done- then it is done,
and all the world wonders why it was not
done centuries ago."
-- Frances Hodgson Burnett
Schrödinger's Equation
Same is true for Newton’s Laws of Motion
Classical Mechanics is the limiting theory
of the more general Quantum Mechanics.
Newton’s Laws of Motion can be derived
from Schrödinger's equation with some
simple assumptions (Ehrenfest theorem).
Therefore Schrödinger's equation is the
governing equation for particles (i.e. m≠0).
Schrödinger's Equation
Many ways to demonstrate plausibility of
Schrödinger's equation.
Easiest to start with a travelling wave in
complex notation:
Ψ(x,t) = A e– i(ωt – kx)
Transform from (ω, k) to (E, p)
Schrödinger's Equation
Transform from (ω, k) to (E, p)
E = hf = ħω [ħ≡h/2π]
k = 2π/λ = 2π(p/h) = p/ħ
ωt – kx = (E/ħ)t – (p/ħ)x = (Et – px)/ħ
Ψ(x,t) = A e– i(ωt – kx)
Ψ(x,t) = A e– (i/ħ)(Et – px)
Schrödinger's Equation
For the wavefunction:
Ψ(x,t) = A e– (i/ħ)(Et – px)
Can extract E and p via derivatives:
∂Ψ(x,t)/∂t = – (iE/ħ)A e– (i/ħ)(Et – px)
∂Ψ(x,t)/∂t = – (iE/ħ) Ψ(x,t)
EΨ(x,t) = iħ∂Ψ(x,t)/∂t
Recall:
i2 = –1  i = –1/i
Schrödinger Equation
For the wavefunction:
Ψ(x,t) = A e– (i/ħ)(Et – px)
∂Ψ(x,t)/∂x = (ip/ħ)A e– (i/ħ)(Et – px)
∂Ψ(x,t)/∂x = (ip/ħ) Ψ(x,t)
pΨ(x,t) = (ħ/i) ∂Ψ(x,t)/∂x
pΨ(x,t) = –iħ ∂Ψ(x,t)/∂x
pΨ(x,t) = –iħ ∂/∂x Ψ(x,t)
Don’t need this now, but save for later
Schrödinger's Equation
Repeating:
∂Ψ(x,t)/∂x = (ip/ħ)A e– (i/ħ)(Et – px)
∂2Ψ(x,t)/∂x2 = (ip/ħ)2A e– (i/ħ)(Et – px)
∂2Ψ(x,t)/∂x2 = –(p/ħ)2 Ψ(x,t)
p2 Ψ(x,t) = –ħ2 ∂2Ψ(x,t)/∂x2
Schrödinger's Equation
Ψ(x,t) = A e– (i/ħ)(Et – px)
Classically for a particle, we expect a
simple result like:
E = K + V = p2/2m + V(x,t)
So it is consistent to multiply by Ψ:
EΨ = p2Ψ/2m + VΨ
Schrödinger's Equation
EΨ = p2Ψ/2m + VΨ
EΨ(x,t) = iħ∂Ψ(x,t)/∂t
p2 Ψ(x,t) = –ħ2 ∂2Ψ(x,t)/∂x2
iħ∂Ψ(x,t)/∂t = –(ħ2/2m) ∂2Ψ(x,t)/∂x2 + V(x,t)Ψ(x,t)
iħ∂Ψ/∂t = –(ħ2/2m)∂2Ψ/∂x2 + VΨ
iħ∂Ψ/∂t = –(ħ2/2m)∇2Ψ+ VΨ
Question
Why wasn’t this a proof?
Schrödinger's Equation
1. We only demonstrated this for a
travelling wave, which is particularly
unlike a particle.
2. We are assuming validity for any
potential V(x,t), i.e. for any force that
leads to that potential.
Schrödinger's Equation cannot be proven
in general.
Brief History
1926: Schrödinger introduces his
equation.
1933: Schrödinger receives Nobel prize
in Physics "for the discovery of new
productive forms of atomic theory"
Schrödinger's Equation
iħ∂Ψ/∂t = –(ħ2/2m)∂2Ψ/∂x2 + VΨ
Schrödinger's Equation is a linear, 2nd order
partial differential equation, a function of Ψ(x,t)
which is a complex scalar, though x can be a
3-d vector. Note that coefficients are not
necessarily constant – i.e. V(x,t).
Schrödinger's Equation is a wave equation –
clear from the way we constructed it from
plane waves.
Schrödinger's Equation
iħ∂Ψ/∂t = –(ħ2/2m)∂2Ψ/∂x2 + VΨ
Other wave equations:
∂2P/∂x2 = (1/v2) ∂2P/∂t2
∇2E = (1/c2)∂2E/∂t2
∇2B = (1/c2)∂2B/∂t2
Schrödinger's Equation
Some similarities to other wave
equations, but differences too:
• Leads to QM version of E = p2/2m + V
• Does not directly lead to fλ = v
• Depends on m, by construction – only
for particles with mass, not photons
• Gives complex wavefunctions, Ψ.
Acceptable because we only measure
|Ψ|2
Schrödinger's Equation
iħ∂Ψ/∂t = –(ħ2/2m)∂2Ψ/∂x2 + VΨ
Schrödinger's Equation is linear in Ψ.
Therefore solutions obey the superposition
principle, as do other wave phenomena.
i.e. if Ψ1 and Ψ2 are solutions, then any
linear combination is also a solution:
Ψ = aΨ1 + bΨ2
Waves
Water waves, on surface of water
Sound waves, in air, liquid, solid
Surface Acoustic Waves (SAW) on
surface of a solid
and many others…
Electromagnetic waves in vacuum or in a
medium.
Waves of What?
What is oscillating in a particle wave??
There is no field corresponding to Ψ like E
and B. Next week will discuss the
‘probabilistic interpretation of the
wavefunction’
Assume a wavefunction, Ψ(x,t), which is a
complex scalar field.
Ψ(x,t) is oscillating. But what is Ψ(x,t) ??
Probability Function
Ψ(x,t) can be positive, negative, and
even a complex number (which is not
in itself measurable).
But |Ψ(x,t)|2 is always a real positive
number, like a wave intensity (which
generally is measurable).
Norm of a complex number
Let Ψ = a + ib
Ψ* = a – ib
a, b real
complex conjugate
|Ψ|2 = Ψ*Ψ
|Ψ|2 = (a – ib)(a + ib)
|Ψ|2 = a2 + b2
Always a real positive number.
Ψ(x,t) is a complex scalar field
Return to Schrödinger’s equation:
iħ∂Ψ/∂t = –(ħ2/2m)∂2Ψ/∂x2 + VΨ
By construction, one solution is:
Ψ(x,t) = A e– (i/ħ)(Et – px)
Note that
Ψ(x,t) = A sin((Et – px)/ħ)
Ψ(x,t) = A cos((Et – px)/ħ)
are not solutions to this wave equation!
Ψ(x,t) is a complex scalar field
Question: does Ψ(x,t) have to be a
complex scalar field?
Quite a hotly debated topic for a very
long time.
See “Quantum Mechanics Must Be
Complex”, posted on Avenue
Standard Argument (1/4)
iħ∂Ψ/∂t = –(ħ2/2m)∂2Ψ/∂x2 + VΨ
Let Ψ = Ψ1 + iΨ2, where Ψ1 and Ψ2 are real
Substitute into Schrödinger's Equation
Separate real and imaginary terms
iħ∂Ψ/∂t = –(ħ2/2m)∂2Ψ/∂x2 + VΨ
iħ∂(Ψ1+iΨ2)/∂t = –(ħ2/2m)∂2(Ψ1+iΨ2)/∂x2 +
V(Ψ1+iΨ2)
Standard Argument (2/4)
Separate real and imaginary terms:
-ħ∂Ψ2/∂t = –(ħ2/2m)∂2Ψ1/∂x2 + VΨ1
iħ∂Ψ1/∂t = –i(ħ2/2m)∂2Ψ2/∂x2 + iVΨ2
-ħ∂Ψ2/∂t = –(ħ2/2m)∂2Ψ1/∂x2 + VΨ1
ħ∂Ψ1/∂t = –(ħ2/2m)∂2Ψ2/∂x2 + VΨ2
What do we learn from this?
Standard Argument (3/4)
Schrödinger's Equation with a complex
Ψ is equivalent to 2 coupled differential
equations for Ψ1 and Ψ2 (similar to E
and B in Maxwell’s equations)
So Ψ did not need to be complex [in this
argument]. It is just a convenient way to
carry along 2 related variables.
Standard Argument (4/4)
Instead we could have worked with its
real and imaginary parts, but they have
no physical significance, unlike E and B.
In the end we only care about
Ψ*Ψ = (Ψ1 + iΨ2) (Ψ1 - iΨ2) = Ψ12 + Ψ22
and we do not need Ψ1 and Ψ2
separately.
So, it is better to work with a complex Ψ
What is ‘waving’ in the wavelike
behaviour of particles?
If you are still wondering, stay tuned for
next week
Conclusion
Particle motions are as described by Newton,
with relativistic corrections by Einstein.
At the same time…
Particles also have a wave-like property,
described by Schrödinger's Equation.
“Wave-particle duality”