2 - States, Observables, And Measurements Flashcards
- Describe the formalism of quantum physics in terms of Hilbert space (complex vector space), linear operators, eigenvalues and eigenvectors, focusing on their formulation as 2x2 matrices for describing two-level systems;
What is |e^(i*phi)|^2 equal to?
1
What are the eigenvalues of operators?
Typically +- hbar/2
What are expectation values?
The average value of repeated experiments. It is calculated by:
=Sigma (probability)*(outcome)
=P_+(+hbar/2)+P_-(-hbar/2)
=hbar/2(|a|^2-|b|^2)
OR
= < Psi | S_n | Psi >
What is an observable?
An observable is simply a measurable quantity. Mathematically, the observable is the operator used.However, observables have the property wherein if you repeat the measurement without the system change in between, then you get the same result.
What is Hilbert Space?
A Hilbert Space, after David Hilbert, is a complex vector space with an infinite-dimensional inner product having the product that it is complete or closed.
What is a spin operator?
A mathematical object that acts or operates on a ket and transforms it into a new ket for example;
A|psi>=|phi>
What are the spin operators for the three cartesian axes?
S_z = hbar/2(1 0;0 -1)
S_y = hbar/2(0 -i;i 0)
S_x = hbar/2(0 1;1 0)
What is the Euler formula?
e^(ix) = cos(x)+i*sin(x)
e^(ix) |^2 = | cos(x)+i*sin(x) |^2 = 1
What is a ket that is not changed by an operator that acts on it?
Eigenvectors of an operator are the only kets that are unaffected by that operator.
What is the eigenvalue equation?
S_n * |+>_n = hbar/2 * |+>_n
Where:
S_n = Operator |+>_n = Eigenvector hbar/2 = Eigenvalue
What is a diagonal matrix?
A matrix having non-zero elements only in the diagonal running from the upper left to the lower right.
An operator is always … in its own basis.
An operator is always diagonal in its own basis.
Eigenvectors are … in their own basis.
Eigenvectors are unit vectors in their own basis.
What is the Born Rule?
The Born rule is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result.
It talks about the probability calculations.
What are state vectors?
Expressions of a particle’s spin
