# Probability Theory and Statistics required for random wave motion analysis

## Summary of Contents

- 1.Probability spaces.
- 2. Random variables
- 3. Lebesgue integral w.r.t. a measure and w.r.t. a probability measure
- 4. Cornerstone theorems of Lebesgue’s theory of integration: Monotone convergence. Fatou’s lemma and dominated convergence.
- 5. Moments of a random variable, mean, variance, skewness and higher moments.
- 6. The direct product of a finite number of measures and Fubini’s theorem on change in the order of integration.
- 7. Karl Pearson’s correlation coefficient.
- 8. Absolute continuity of measures, the Radon-Nikodym theorem and probability distributions and densities.
- 9. Conditional expectation and conditional probability.
- 10. Quantum probability space: Events, observables and states in a quantum probability space.
- 11. Joint probability distribution of classical random variables.
- 12. Heisenberg’s uncertainty principle for non-commuting observables and the impossibility of defining joint probabilities for non-commuting observables in the quantum theory.
- 13. Positive definitivity of the joint characteristic function of classical random variables.
- 14. Bochner’s theorem in classical probability.
- 15. Non-positive definitivity of joint characteristic functions of non-commuting observables in the quantum theory.
- 16. The inequalities of John Bell and the consequent impossibility of constructing a hidden variable theory for quantum mechanics.

- 17. Classical stochastic processes: Brownian motion, Poisson process and more generally Levy processes.
- 18.Ito’s formula for Brownian motion and Poisson processes.
- 19. Quantum Brownian motion and quantum Poisson processes: Noncom-mutative generalizations of classical BM and classical PP.
- 20. Realizing classical stochastic processes using families of operators in Boson Fock space in a coherent state.
- 21. Bernoulli random variables in classical and quantum probability.
- 22. Stochastic differential equations driven by Brownian motion: Existence and Uniqueness.
- 23. Stochastic differential equations driven by Levy processes.
- 24. The generalized Ito-Doleans Dade-Meyer differential rule for discontinuous Martingales.
- 25. Girsanov’s formula for the change of drift in an sde.
- 26. Quantum stochastic differential equations in the sense of Hudson and Parthasarathy; Existence and Uniqueness of solutions.
- 27. Some practical applications of quantum probability.

[a] Computation of scattering cross sections.

[b] The wave function of our universe: Wheeler-Dewitt-Hartle-Hawking’s theory. The Hartle-Hawking theory of the probability distribution of the radius of our universe with a scalar field coupling.

[c] Computation of tunneling probabilities of a quantum particle through a potential barrier with applications to tunneling diode.

[d] Probability of induced transitions in a laser; The interaction of Light with atoms-The Glauber-Sudarshan theory.

[e] Energy bands in a semiconductor-the theory based on Bloch wave functions; Specialization to the Kronig-Penney model.

[f] Modeling quantum systems coupled to a noisy bath—The Hudson-Parthasarathy equation and its solution using Maasen’s kernel approach and using the functional form of the Glauber-Sudarshan representation.

## Probability spaces and measure theoretic theorems on probability spaces

A triplet (Q, *J-, P)* where U is called the sample space of the experiment, *P* is the a-field of events and P is a probability measure.

- 2. Random variables
- 3. Lebesgue integral w.r.t. a measure and w.r.t. a probability measure
- 4. Cornerstone theorems of Lebesgue’s theory of integration: Monotone convergence, Fatou’s lemma and dominated convergence.
- 5. Moments of a random variable, mean, variance, skewness and higher mo-ments.
- 6. The direct product of a finite number of measures and Fubini’s theorem on change in the order of integration.
- 7. Karl Pearson’s correlation coefficient.
- 8. Absolute continuity of measures, the Radon-Nikodym theorem and probability distributions and densities.
- 9. Conditional expectation and conditional probability.

## Basic facts about quantum probability

- 10. Quantum probability space: Events, observables and states in a quantum probability space.
- 11. Joint probability distribution of classical random variables.
- 12. Heisenberg’s uncertainty principle for non-commuting observables and the impossibility of defining joint probabilities for non-commuting observables in the quantum theory.
- 13. Positive definitivity of the joint characteristic function of classical random variables.
- 14. Bochner’s theorem in classical probability.
- 15. Non-positive definitivity of joint characteristic functions of non-commuting observables in the quantum theory.
- 16. The inequalities of John Bell and the consequent impossibility of constructing a hidden variable theory for quantum mechanics.

Given three random variables *Xi,* X_{2}, ^3, each assuming values ±1 only, we observe that

*X _{3}(X!-X_{2})* < l-XiX

_{2}

Taking expectations gives

E(X!X_{3}) - E(X_{2}X_{3}) < 1 - E(%!X_{2})

Interchanging *Xi* and *X _{2}* gives

E(X_{2}X_{3}) - E(XrX_{3}) < 1 - E(X1X_{2})

Thus,

lEpQXa) - E(X_{2}X_{3})I < 1 - E(XjX_{2})

This is called Bell’s inequality for Bernoulli random variables. It is violated in certain cases for quantum Bernoulli random variables. For example, consider _{2},n_{3} and consider the observables

*Xk = (nie, a) = ntiCTi* + n_{fc2}_{3}<7_{3}, *k* = 1, 2,3

Then,

*XkX _{m} = (nk,n_{m}) +i(a,hk* x n

_{m})

Now, we can choose the state

*P =* (W_{2}

Then,

Tr(p<7fc) = 0, *k* = 1,2,3

and we have

*Tr{pXkX _{m}) = (hk, hm), k, m* = 1, 2,3

Let *Okm* be the angle between *hk* and *n _{m}.* Then to verify Bell’s inequality, for the quantum observables

*Xi, X%, X3,*we require that

|cos(0i_{3}) - COS(6*23)| < 1 - cos(0i_{2})

for all points on the unit sphere. Now, suppose we choose n_{3} as the north pole and |#i — *O2*1 < e. Then by an appropriate choice of e, Bell’s inequality will be violated.