Quantum Mechanics

1. Superposition

Quantum state is a (complex) superposition of classical states.

Here, the coefficients are complex numbers called amplitudes.

The states

are the basis of the N-dimensional Hilbert space.

For example, the state can be spin state, angular momentum, etc.

2. Measurement

The (superposed) state is not observable, but the we will see only one of the classical state by "measurement". The probability of measurement to a state, |j>, is the square of the amplitude.

$$ P(j) = |\alpha_j|^2 $$

The measurement is non-unitary, i.e., irreversible.

3. Unitary Evolution

Operator: change a state to another state