Quantum Mechanics is a
Mathematical Model
Planck’s constant/2π  10-34 Joule-sec
• One assumes:
These can not be derived
(1) energy,
E  (-ℏ/i)/t
-- they are postulates!
t = time
(2) momentum, P  (ℏ/i)
(3) particle probability density, (r,t)
The gradient operator
 = i/x + j/y + k/z
 = 2 = 2/x + 2/y2 + 2/z2
“deriving” the Schrodinger Equation
In classical physics we write kinetic energy as
T = ½ mv2
= (mv)2/2m = P2/2m
P = mv = momentum
Using P  (ℏ/i)
(QM assumption)
(1/2m) P  P  (1/2m)(ℏ/i)2  
T  (-ℏ2/2m)2
Schrodinger Equation for
Hydrogen atom
E = kinetic energy + potential energy
Coulomb
potential energy
(-ℏ/i) /t
= (-ℏ2/2m)2 - e2/r
Finally, these operators act on (r,t)!
Schrodinger Equation for H atom
[(-ℏ22 /2m)2 - e 2/r ] (r,t) = iℏ /t (r,t)
• The physics in this equation is not easily visualized.
*(r,t) (r,t) dV represents
the probability that the electron in the hydrogen atom
can be found within a volume, dV= dxdydz, at (r,t).
Since the electron must be somewhere,
 * dxdydz = 1
This turns out to be very important!
Hydrogen Schrodinger Equation time dependence
[(-ℏ22 /2m)2 - e 2/r ] (r)e –iEt/ℏ = E (r) e –iEt/ℏ
… time dependence is exponential, E = constant
(-ℏ/i) /t e –iEt/ℏ = E e –iEt/ℏ
• Model predicts experimental atomic structure
observed in the laboratory (model is correct!)
• Model implies that an electron behaves like a
wave when it is confined to 10-8 cm distances.
• All one needs is differential equations to solve
for (r)!
With the time dependence factored out
[(-ℏ22 /2m)2 - e 2/r ] E (r) = E  E(r)
… it’s an eigenvalue equation!
A strange outcome is that the mathematical boundary
condition on ,
 * dxdydz = 1
limits the allowed values for E.
Quantization of Energy!
Quantization is a
mathematical result!
Ensuring that the
integral does not
diverge is not easy!
1 eV = 1.6 x 10-19 Joule
n = 1, 2, 3, …
Old classical model for the hydrogen atom
The mathematics helps us describe
and quantify this “electron cloud”.
Details!
All this comes from
requiring that the
Integral converge.
Everything that we know is made of atoms: the
mathematics that determines the structure of atoms
and the molecules formed from them is crucial to all
of chemistry, biology and materials science!
ICE
Graphene
Cosmology … and such
Those who study the smallest particles – electrons,
quarks, neutrinos and the rest of the basic building
blocks of our world have an extraordinary story -about how our universe developed from the big bang.
The story is based on mathematics.
First of all, they start with a kind of Schrodinger equation
model – one which is appropriate for particles which are
moving very fast. It is called Quantum Field Theory. In this
field theory there is a field operator, (r,t), for the electron.
Then they do something rather startling: they postulate that the
laws of physics (their equations) should be invariant under a kind
of “rotation”, called a gauge transformation.
They discovered that it won’t work unless they have a photon
(called a gauge particle) to “undo” the rotation.
1. Initial state
2. Rotate 

Aµ

’
3. Transform A
’
Aµ
4. Final state
invariance

Note that the photon field
must also be transformed.
The photon
“undoes” the
rotation and
preserves the
symmetry!
These symmetries are called Gauge Symmetries. In putting this idea into a
mathematical formalism they found that the exact formula for how the extra
(photon-like) particles interact with the electrons and quarks fell out of the their
“invariant” equations. Math again gives the answers!
From this approach (and a lot of sophisticated mathematics) came the
Standard Model of particle physics – the “theory of everything”.
HOWEVER, the symmetry is broken when the photon-like particles take
on mass. Not to waste an unexpected result, the particle physicists used
this symmetry breaking to describe how the universe evolved from the Big
Bang!
The idea is this: as the universe cools it “condenses” out the gauge particles
(they take on mass) and the symmetries are broken. From the equations,
the particle physicists have been able to develop a step by step scenario
describing how the universe as we know it emerged from the big bang . Many
questions remain.
Thank you!
From this field operator a kind energy operator, called a Lagrangian density,
L, is constructed. A similar operator in classical physics is known to reproduce
Newton’s law, F= ma, when one minimizes Ldt over the path of the motion.
Then the particle physicists do a very interesting thing:
They demand that this Lagrangian energy operator be invariant under
a special mathematical operation, called a gauge transformation.
The gauge transformation is a kind of rotation which changes the complex
phase of the electron by an arbitrary function which depends on where
(in space-time) the particle is.
Cosmology … and such
Those who study the smallest particles – electrons, quarks, neutrinos and
the rest of the basic building blocks of our world have an extraordinary
story -- about how our universe developed from the big bang.
The story is based on mathematics.
First of all, they start with a kind of Schrodinger equation model – one which
is appropriate for particles which are moving very fast. It is called Quantum
Field Theory. In this field theory there is a field operator, (r,t), for the
electron. The field
operator is a linear combination of all possible free particle states for an
electron. With this field operator one can create (or destroy) a free electron
at any point in space, and with any energy – if the math calls for it.
What they find is that the invariance can not occur unless one introduces
another field, called the gauge field. This gauge field emerges as the force
field by which electrons interact with each other: the photon field!
In other word, the gauge “SYMMETRY” demands that the photon exists!
Furthermore, the modified Lagrangian (wth the photon field) prescribes
exactly how the electron interacts with the photon!
This gave them the idea to look at the expanding, cooling universe as a
series of symmetries which are broken as particle fields “Crystallize out”.
It has been so successful that it is now the “accepted” story of how the
universe started from a small dense, hot system to the vast array of
galaxies and stars as we know it today.
ℓ = 2, m = -2,-1, 0, 1, 2