1. Applications of Mathematics to Science and Engineering* Dr. Barbara Hale Physics Department Missouri S&T * talk given to the Student Chapter of MAA Missouri S&T October 5, 2011

2. 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.

3. 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 = mc 2

4. 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 /  y 2 +  2 /  z 2 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

5. In classical physics we write kinetic energy as T = ½ mv 2 = (mv) 2 /2m = P 2 /2m Using P  ( ℏ /i)  (QM assumption) (1/2m) P  P  (1/2m) ( ℏ /i) 2    T  (- ℏ 2 /2m )  2 “deriving” the Schrodinger Equation P = mv = momentum

6. E = kinetic energy + potential energy (- ℏ /i)  /  t = (- ℏ 2 /2m )  2 - e 2 /r Schrodinger Equation for Hydrogen atom Coulomb potential energy Finally, these operators act on  (r,t)!

7. 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! [(- ℏ 2  2 /2m )  2 - e 2 /r ]  (r,t) = i ℏ  /  t  (r,t)

8. 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 [(- ℏ 2  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

9. 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! [(- ℏ 2  2 /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

10. n = 1, 2, 3, … 1 eV = 1.6 x 10 -19 Joule

11. Old classical model for the hydrogen atom The mathematics helps us describe and quantify this “electron cloud”.

12. Details! All this comes from requiring that the Integral converge.

13. 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

14. Graphene

15. 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!

16. 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 mathematics we 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.

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

19. 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.

20. 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.

21.  ’’ AµAµ AµAµ  ’’  invariance Note that the photon field must also be transformed. 1. Initial state 2. Rotate  3. Transform A4. Final state The photon “undoes” the rotation and preserves the symmetry!

22. 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.

25. Thank you!

26. 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.

27. 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.

28. 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 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 !

29. ℓ = 2, m = -2,-1, 0, 1, 2