postulates of nonrelativistic quantum mechanics.

Classical Mechanics

1. The state of a particle at any given time is specified by the two varibles $x(t),p(t)$.[phase space]

2. Every dynamical varible $\omega$ is a function of $x,p$

3. If the particle is in a state given by $x,p$, the measurement of the varible $\omega$ will yield a value $\omega$. The state will remain unaffected.[ideal measurement]

4. Hamilton’s canonical equations

   $$ \dot x=\frac {\partial \mathcal H}{\partial p}\\\dot p=-\frac {\partial \mathcal H}{\partial x} $$

Quantum Mechanics

1. The state of the particle is represented by a vector $|\psi(t)\rangle$ in a [Hilbert space].

2. The independent varibles are represented by Hermitian with $[\hat x,\hat p]=i\hbar$.The operators corresponding to dynamical varibles $\omega$ are given Hermitian operators$\Omega(\hat x,\hat p)$

3. $|\psi\rangle$,$P(\omega)\propto|\lang \omega|\psi\rang|^2$ .The state of the system will change from $|\psi\rang$ to $|\omega\rang$ as a result of the measurement.

4. Schrodinger equation

   $$ i\hbar\frac {\partial }{\partial t}|\psi(t)\rang=H|\psi(t)\rang $$

[eigenvalue spectrum] wave function in the $\omega$ space: $\psi(\omega)=\lang\omega|\psi\rang$

$$ |\psi\rang=\int|\omega\rang\lang\omega|\psi\rang d\omega $$

[ideal quantum measurement]an ideal measurement of any varibles in classical mechanics leaves any state in variant, whereas the ideal measurement of $\Omega$in quantumn mechanics leaves only the eigenstate of $\Omega$ invariant

[quantum ensemble] which consists of a large number $N$ of particles all in the same state $|\omega\rang$ .$|\omega\rang=a_1|1\rang+...$, For sufficiently large $N$, the deviations from $a_1$ will be negligible. Classcial ensemble: properties before measurement| quantum ensemble: states after measurement [ONLY differnece is measurement&determination, quntum need ensemble to determine initial state but classic could calculate.]

[statistics of a state]$\lang\Omega\rang=\sum P(\omega_i)\omega_i=\sum_i|\lang\omega_i|\psi\rang|^2\omega_i=\sum\lang\psi|\omega_i|\omega_i\rang\lang\omega_i|\psi\rang=\lang\psi|\Omega|\psi\rang$ we mean the average over the ensemble corresbonding to the state.

[compatible & incompatible variables]measurement above→single filter process; multiple filtering process: after $\Omega$ we immediately measure $\Lambda$ ; degenerate case;……………..

$$ [\Omega,\Lambda]=0 $$

[density matrix] ensemble of $N$ systems all in the same state→ ensembles of $N$ systems, $n_i$ of which are in the state $|i\rang$, thus the ensemble is described by k kets and k occupabcy numbers. A convenient way to assemble all this information is in the form of the density matrix

$$ \rho=\sum_ip_i|i\rang\lang i|. $$

The pure state(ensemble) correpond to all $p_i=0$ except one.

the mean value for ensemble is $\lang \overline \Omega\rang=\sum p_i\lang i|\Omega|i\rang\to Tr(\Omega \rho)$