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, X2, ^3, each assuming values ±1 only, we observe that
X3(X!-X2) < l-XiX2
Taking expectations gives
E(X!X3) - E(X2X3) < 1 - E(%!X2)
Interchanging Xi and X2 gives
E(X2X3) - E(XrX3) < 1 - E(X1X2)
Thus,
lEpQXa) - E(X2X3)I < 1 - E(XjX2)
This is called Bell’s inequality for Bernoulli random variables. It is violated in certain cases for quantum Bernoulli random variables. For example, consider
Xk = (nie, a) = ntiCTi + nfc2
Then,
XkXm = (nk,nm) +i(a,hk x nm)
Now, we can choose the state
P = (W2
Then,
Tr(p<7fc) = 0, k = 1,2,3
and we have
Tr{pXkXm) = (hk, hm), k, m = 1, 2,3
Let Okm be the angle between hk and nm. Then to verify Bell’s inequality, for the quantum observables Xi, X%, X3, we require that
|cos(0i3) - COS(6*23)| < 1 - cos(0i2)
for all points on the unit sphere. Now, suppose we choose n3 as the north pole and |#i — O21 < e. Then by an appropriate choice of e, Bell’s inequality will be violated.
