Applications of indeterminacy
(Quantum random number generators)
Fundamentally random
(Quantum cryptography)
Bennett Brassard 84 scheme.
Steps for quantum key distribution:
(1) Alice and Bob choose
randomly (they use quantum
random number generator).
(2
Neutrino oscillation is an example of spin precession in a three level
system.
Operator time evolution.
Spin precesses similar to Bloch vector.
Trivial example of a time dependent Hamiltonian.
Adiabatic spin dynamics and Berry phase.
Where
is a small para
Measurement and entanglement (Schrodingers cat):
Wave function collapse:
The observer can then make a table for each particle. But from previous
experience she knows that :
if
then
if
then
The probability of getting
is given by
This is identical to the de
Quantifying randomness and entanglement: Von Neumann entropy.
Classical Boltzmann entropy S is dened as
If we had a pair of separated systems U and V so that the joint probability
is a product (i.e. they are independent) or equivalently their joint densi
Schrodingers equation for the unitary operator
Dening the time evolved state
we see that
This is the Schrodinger equation for time evolution of states.
Time evolution of density matrices is analogous.
Time evolution in Bloch sphere.
Consider a two level
Physics 8.06
Apr 1, 2008
SUPPLEMENTARY NOTES ON THE
CONNECTION FORMULAE FOR THE
SEMICLASSICAL APPROXIMATION
c R. L. Jae 2002
The WKB connection formulas allow one to continue semiclassical solutions from an allowed to a forbidden region and vice versa. Ho
Excursion into Pauli matrices (derivation assigned as HW) :
Hermitean matrices form a real vector space.
If the density matrix is expanded similarly then
The Bloch sphere.
Since for density matrices
It follows that
Furthermore since
We obtain that
For tw
Example: Fully mixed (unpolarized state)
is basis independent i.e.
The expectation values are also rotation invariant.
Excursion into Pauli matrices (derivation assigned as HW) :
Hermitean matrices form a real vector space.
If the density matrix is expand
Compatible observables.
Consider the following SG scenario:
Similar to classical logic, eigenstates of compatible observables are
labelled by eigenvalues of both observables.
Observables with simultaneous eigenvalues commute. A and B are
diagonal matrice
Phys622 Lecture 2 09/04/13
axioms of quantum mechanics
(1) State
Vectors
(Kets)
(2) Inner product between two states:
such that
Inner product is linear
Probability has a convenient interpretation if:
so that
(3) Allows denition of dual vector (bra)
(4)
Unitary invariant functions of operators.
Dening
in any basis
To see why Tr(A) is unitary invariant we use the spectral decomposition of
A
fact that
so that
is basis independent. Tr is characterized by the property
Any function f(A) denes more basis
Suppose the Hamiltonian is time-independent. Then the Schrodinger
equation has the form
Expanding
in an eigenbasis of H
Therefore energy eigenstates
Evolve into themselves
.
Eigenstates
of a commuting observable
are also eigenstates of energy and theref