Applications of Mathematics to Science and Engineering*

Dr. Barbara HalePhysics Department

Missouri S&T

* talk given to the Student Chapter of MAA Missouri S&T October 5, 2011

The Role Mathematics Plays in Science and Engineering

• Models of physics, chemistry, biology and engineering are not easily understood using words and pictures.

• Mathematics provides the language in which these models are formulated, understood, and applied to real life situations.

A few examples

• Structure of the atom/chemistry/materials:

quantum mechanics• Wave propagation (light, radio waves, xrays):

Maxwell’s equations• Cosmology / planetary motion:

Newton’s laws, special relativity, E = mc2

• One assumes: (1) energy, E (-ℏ/i)/t (2) momentum, P (ℏ/i) (3) particle probability density, (r,t)

= i/x + j/y + k/z = 2 = 2/x + 2/y2 + 2/z2

Quantum Mechanics is a Mathematical Model

These can not be derived -- they are postulates! t = time

The gradient operator

Planck’s constant/2π 10-34 Joule-sec

In classical physics we write kinetic energy as

T = ½ mv2 = (mv)2/2m = P2/2m

Using P (ℏ/i) (QM assumption)

(1/2m) P P (1/2m)(ℏ/i)2 T (-ℏ2/2m)2

“deriving” the Schrodinger Equation

P = mv = momentum

E = kinetic energy + potential energy

(-ℏ/i) /t = (-ℏ2/2m)2 - e2/r

Schrodinger Equation forHydrogen atom

Coulomb potential energy

Finally, these operators act on (r,t)!

Schrodinger Equation for H atom

• 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!

[(-ℏ22 /2m)2 - e 2/r ] (r,t) = iℏ /t (r,t)

• 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)!

Hydrogen Schrodinger Equation time dependence

[(-ℏ22 /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

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!

[(-ℏ22 /2m)2 - e 2/r ] E (r) = E E(r) … it’s an eigenvalue equation!

Ensuring that the integral does not diverge is not easy!

With the time dependence factored out

n = 1, 2, 3, …

1 eV = 1.6 x 10-19 Joule

Old classical model for the hydrogen atom

The mathematics helps us describe and quantify this “electron cloud”.

Details!

All this comes fromrequiring 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

• A famous mathematical theorem by Helmholtz:

To determine a vector field, E(r,t), you need only

specify (everywhere) its

a) divergence E =

b) curl x E = J(r,t)

Maxwell’s Equations for electric field, E and magnetic field, B

This is math!

Maxwell’s Equations

E =

x E = - B/t B = 0

X B = - E/t + J

charge density – source of E

moving charges - source of B

E and B are coupled!!

This is a miraculous combination of mathematics and physics. We know E and B are coupled only because laboratory experiments tell us so. But without the mathematicswe would be unable to predict the complicated phenomena Maxwell’s equations describe !The model is relativistically correct – electromagnetic waves travel with the speed of light.

Canberra 70m (230 ft.) antenna -- one of the three complexes which comprise NASA's Deep Space Network.

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.

’

AµAµ

’

invariance

Note that the photon field must also be transformed.

1. Initial state 2. Rotate

3. Transform A 4. Final state The photon“undoes” therotation andpreserves thesymmetry!

These symmetries are called Gauge Symmetries. In putting this idea into amathematical 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 theStandard 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.

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.

What they find is that the invariance can not occur unless one introducesanother field, called the gauge field. This gauge field emerges as the forcefield 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!

ℓ = 2, m = -2,-1, 0, 1, 2