Quantum mechanics

Pure quantum states

Discrete states as vectors

Observables as linear operators

Orthonormal basis

Constructing a Hermitian matrix for an observable

Spin of a single particle

Mixed quantum states

Mixed quantum states

Probability amplitudes


State evolution

Indexing states to time

We have state defined at each time \(t\).

\(\Psi (t)\).

Wave functions

We have state \(\Psi (t)\).

\(\psi(x,t)=\langle x\rvert \Psi (t) \rangle\)

This is the wave function.


Discrete time

With discrete time we can use a canonical operator for moving between discrete states in single jumps.

With discrete time there must a countable number of states.

We can index time to the integers.

At time \(0\) we have \(v\)

At time \(1\) we have \(\Psi v\)

At time \(2\) we have \(\Psi \Psi v\)

We can write this as \(\Psi (t_1,t_0)=\Psi ^{t_1-t_0}\)

Representation theory for the time group

Time is a linear operator

Instead, we describe the time operator as a Lie group, using Lie algebra.

\(\Psi (t_b-t_a)=e^{(t_b-t_a)X}\)

States are vectors

We can remove a degree of freedom by using norm of 1 for vectors

For each dynamic system we define a set of possible states.

We can describe a state \(v\in V\).

Finite state spaces

We can describe a system like heads or tails.

Infinite state spaces

This can describe continous position, or an angle.

Indexing time to the real numbers

Sloan’s theorem

Continous time with Lie algebra

We use \(X=iH\), what are the implications of this compared to other choices?

Lie algebras with \(n\times x\)

This loops back? multiple dimensions, infinite, so maybe not?

With continuous time we do not have a single operator to describe movements. There is always one smaller.

With continous time there must be either a single state, or an uncountably infinite number of states.



\(U=\lim_{n\rightarrow \infty }(I+\dfrac{1}{n}G)^n\)









\(H\) is Hermitian

\(U=\lim_{n\rightarrow \infty }(I+\dfrac{1}{n}iH)^n\)

This isn’t quite right, need defined for different time jumps.

Unitary time

Why? What’s the interpretation here? Is this an assumption, or just a modelling choice?

\(\Psi (t_b-t_a)^* \Psi (t_b-t_a)=e^{(t_b-t_a)X^*} e^{(t_b-t_a)X}\)

\(\Psi (t_b-t_a)^* \Psi (t_b-t_a)=e^{(t_b-t_a)(X^*+X)}\)


\(\Psi (t_b-t_a)^* \Psi (t_b-t_a)=e^{(t_b-t_a)(-iH+iH)}=I\)

\(\Psi (t_b-t_a)=e^{(t_b-t_a)iH}\)

The time-depedendent general Schrödinger equation


\(v(t+\delta )=e^{\delta X}v(t)\)

\(v(t+\delta )=(I+\delta X)v(t)\)

\(\dfrac{v(t+\delta )-v(t)}{\delta }=Xv(t)\)

\(\dfrac{\delta v(t)}{\delta t}=Xv(t)\)

\(\dfrac{\delta v(t)}{\delta t}=iHv(t)\)

The energy operator and the time-indepedendent general Schrödinger equation

\(E=ih\dfrac{\delta }{\delta t}\)


Infinite dimensional quantum states




Moving to 3 dimensions

The action integral


Quantum entanglement


The Hamiltonian of quantum mechanics

Plank’s constant

We can add Plank’s constant, due to the arbitrary scaling of time.

Phase shift

Density matrix

Born rule

Spin-statistics theorem

Heisenberg’s uncertainty principle

Result of spin-statistics theorem?

The Dirac equation

The quantum harmonic oscillator