Quantum Measurements

General definition

Let be a Hilbert Space.
Consider a collection of Linear Operators with 𝟙
We say are quantum measurements if for any state :

1. the probability that occurs as the result of mmt is
2. the state post mmt is

Projective measurement

Quantum measurements which are orthogonal projections are called projective mmts
(i.e. they are Hermetian and ).
Now any Hermetian operator can be represented as

where projected spaces of correspond to eigenspaces of with eigenvalue
This defines the Observables
Note that now the possible mmt results correspond to eigenvalues of the observable

Positive operator valued measure (POVM)

Consider a mmt .
Then we define positive semi-Definite operators:

This family of operators is called a POVM.
We can always recover from a POVM by taking roots (as they are positive).

Part IB/II

Let be a complete orthogonal set of wavefunctions, meaning and any wavefunction can be written as:

Then it follows that . More generally, .

• Any observable has associated Hermetian operator .
• Hence, for any observable, we can write as a linear combination of eigenfunctions of .
• The outcome of a measurement is always an eigenvalue of .
• After a measurement with outcome , the wavefunction collapses to a linear combination of eigenfunctions with eigenvalue
• The probability of an outcome is (or the sum of appropriate stuff)
• The expected value of is

Examples

• Position operator is
• Momentum operator is
• Energy operator is
• Angular momentum operator is

Commutator

Composite system

Suppose we have with basis
and we want to only measure in the basis
We take
where

and take the projective measurements in these subspaces and etc whatever …