Axioms Of Quantum Mechanics

In terms of State Vectors¶

The following axioms are for closed quantum systems, hence, all systems are in pure states and all dynamics are unitary. Measurement is only considered via observables and not extended to generalised measurements.

1. States - The state of any physical systems is represented by normalised vector in a complex Hilbert space, $\ket{\psi}~\in~\mathcal{H}$, where $\braket{\psi|\psi} = 1$.

2. Evolution - closed quantum systems evolve via unitary operators

3. Measurement - Observables are given by Hermitian operators. For the observable $O = \sum \lambda_k \ket{\lambda_k}\bra{\lambda_k}$:
   • The probability of measuring $O$ and getting the outcome $\lambda_k$ is
   $$\textrm{Prob}(\lambda_k) = \vert \braket{\lambda_k|\psi} \vert^{2}$$

   (2)
   • The expectation value of $O$ is given by
   $$\braket{O} = \bra{\psi} O \ket{\psi} = \sum_k \lambda_k \vert \braket{\lambda_k|\psi} \vert^{2}$$

   (3)
   • The state after measuring $O$ and getting the measurement outcome $\lambda_k$ is given by
   $$\ket{\psi_{post}} = \frac{ \ket{\lambda_k}\braket{\lambda_k|\psi}}{ \sqrt{ \vert \braket{\lambda_k|\psi} \vert^{2} }}$$

   (4)

In terms of Density Operators¶

These axioms cover quantum systems where the state has interacted with an environment to become mixed, but now evolves via closed dynamics. Measurement in only considered via observables and not extended to generalised measurements.

1. States - The state of any physical systems is represented by a density operator $\rho$, where $\rho \geq 0, ~ \textrm{tr} \big[ \rho \big] = 1$.

2. Evolution - closed quantum systems evolve via unitary operators

3. Measurement - Observables are given by Hermitian operators. For the observable $O = \sum \lambda_k \ket{\lambda_k}\bra{\lambda_k}$:
   • The probability of measuring $O$ and getting the outcome $\lambda_k$ is
   $$\textrm{Prob}(\lambda_k) = \textrm{tr}\big[ \rho \ket{\lambda_k}\bra{\lambda_k} \big]$$

   (6)
   • The expectation value of $O$ is given by
   $$\braket{O} = \textrm{tr} \big[ \rho O \big]$$

   (7)
   • The state after measuring $O$ and getting the measurement outcome $\lambda_k$ is given by
   $$\rho_{post} = \frac{ \ket{\lambda_k}\bra{\lambda_k} \rho \ket{\lambda_k}\bra{\lambda_k} }{\textrm{tr}\big[ \rho \ket{\lambda_k}\bra{\lambda_k} \big]}$$

   (8)