Quantum Mechanics 🏡

Quantum mechanics is the the foundation of all the quantum sciences, most notably QFT.

Contents

Dirac-von Neumann axioms

The space of states is a complex (countably infinite) Hilbert space $\mathbb{H}$. The following axioms apply to the qm system:

  1. The observables are defined as self-adjoint operators on $\mathbb{H}$
  2. The state $\psi$ is a unit vector of $\mathbb{H}$
  3. The expectation value of an observable $A$ is $⟨\psi,A\psi⟩$

From these principles it is possible to derive the Schrödinger equation.

To do this one uses the unitarity of the time evolution operator and first-order expansions thereof.

Dirac–von Neumann axioms - Wikipedia

Postulates of quantum mechanics

Add spin and the exclusion principle (particle statistics) for a “complete” description.

Mathematical formulation of quantum mechanics - Wikipedia

Spin

Intrinsic angular momentrum. No parallel in classical physics. Simplest quantum mechanical system that shows the essence of quantum nature/logic of quantum systems.

Exclusion principle

Pauli’s exclusion principle says that only one fermion can occupy any given quantum state at a time. Ie. the result of exchange of identical particles is different depending on the constituent particle’s spin:

$$\psi(…,\bold{r}_i,\sigma_i,…,\bold{r}_j,\sigma_j,…) = (-1)^{2S}\psi(…,\bold{r}_j, \sigma_j,…,\bold{r}_i,\sigma_i,…)$$

That is under interchange of identical particles we have:

For fermions the condition that the wave function is antisymmetric means that for two fermions in the same state, exchange of the two particles should change sign. Since the particles themselves are identical however the wave function should not change at all. The only way for this to make sense; remain identical and change sign, is for the wave function to be 0 everywhere, ergo the state does not exist.

Time dependence pictures

  1. Schrödinger picture - states carry time dependence
  2. Interaction (Dirac) picture - states and operators carry time dependence
  3. Heisenberg picture - operators carry time dependence

Schrödinger picture

The unitary time evolution operator is responsible for evolving the states in time.

$$ \ket{\psi(t)} = \hat{U}(t,t_0) \ket{\psi(t_0)} $$

This in turn leads to the Schrödinger equation:

$$ i\hbar\frac{\partial}{\partial t} \ket{\psi(t)} = \hat{H}\ket{\psi(t)} $$

Heisenberg picture

The operators here change with time through a unitary transformation according to: $$ A(t) = U^{\dagger}(t)AU(t) $$

The Hesenbergian Hamiltonian is thus $$ H(t) = U^{\dagger}(t)HU(t) $$

Using the change of $U(t)$ this yields

$$ \frac{d}{dt}A(t) = \frac{i}{\hbar}[H,A(t)] + \frac{\partial A}{\partial t} \bigg\vert_{H} $$

The Heisenberg equation of time dependence.

Dirac picture

The time dependence is divided between the states and the operators. Hence there are two Hamiltonians; the free Hamiltonian for observables and the interaction Hamiltonian for states.

In practise the Hamiltonian is broken in two and then a term is multiplied into the states and the operators to remove the free Hamiltonian from the states and the intaraction Hamiltonian from the operators.

This picture does not exist in general for QFT, see Haag’s theorem - Wikipedia. (Advanced stuff)

Schrödinger equation

TODO: Derive from the Dirac-von Neumann axioms

Schrödinger equation

States

As mentioned above the space of states in QM is a complex Hilbert space $\mathbb{H}$. The dimension of this space depends on the system in consideration but is in general infinite.

An important part of any Hilbert space is the inner product it comes with. This is due to the fact that a Hilbert space is also per definition a complete metric space.

$$ ⟨\psi, \phi⟩ \in \Z $$ $$ ⟨\psi, \phi⟩ = ⟨\phi, \psi⟩^* $$ $$\lVert \psi \rVert = \sqrt{⟨\psi, \psi⟩} \in \reals$$

Mathematicians generally assume linearity in the first argument and cojugate linearity in the second. The opposite is the case in physics by convention.

Orthonormality and bases

Using the inner product we can define angles and norms, and thus orthonormality:

$$ ⟨e_i, e_j⟩ = \delta_{ij} $$

where the $e_i$ are an orthonormal basis Using this basis we can define components $\psi_i$ of arbitrary elements $\psi$ in $\mathbb{H}$:

$$ \psi = \sum_i ⟨\psi, e_i⟩e_i = \sum_i \psi_i e_i $$

Physical states, phase factors

An element in state space can naturally be multiplied by any complex number $c$. The question then is, does this new element correspond to a new physical state?

The answer is no. Because of the interpretation of the inner product as probabilities the norm of any physically meaningful state must be normalised to 1. This still leaves one degree of freedom though as the normalisation only restricts one of the two degrees that complex multiplication allows for. Expressing c in polar form:

$$c = r e^{(i\theta)}$$

The normalisation condition effectively removes r (no matter what it is the resultant state is re-normalised) and we are left with whats called the phase factor. This factor does in itself not carry any physickl meaning, but the different in phase between states can sometimes have important consequences.

The distinction between elements of $\mathbb{H}$ and physical states is often overlooked in introductory litterature.

Rays (Projective Hilbert spaces)

The above leads us to define states as rays (one dimensional projective spaces) in $\mathbb{H}$ rather than elements.

Rays are the set of equivalence classes of elements $\psi \neq 0 $ for the relation:

$$ \psi \backsim \phi \rArr \exists \lambda \neq 0, \psi = \lambda\phi$$

Operators and measurments

Having constructed a suitable space of states for our system the next thing we have to do is figure out how to represent observables.

That is how we represent the various measurable quantities of our system, the possible values of measurment and the resultant states.

Linear, Hermitian (self-adjoint) operators from $\mathbb{H}$ to itself fit the bill because:

There are some complications regarding unbounded operators because of the difference between symmetric (Hermitian) and self-adjoint operators in this case: unbound operators do not necessarily have eigenvectors that constitute an orthonormal basis.

Measurments

The results of measuring the observable quantity represented by an operator $A$ are its eigenvalues. The resulting state after measurment is the corresponding eigenvectors:

$$ A\psi_i = \lambda_i\psi_i $$

Components

The operators can be expanded as a sum

$$ A = \sum ⟨A e_i, e_j⟩e_i \otimes e_j = \sum A_{ij}e_{ij} $$

Where the $A_{ij}$ are its components; matrix elements.

Dirac bra-ket notation

Because of the natural isomorphism that exists between many of the Hilbert spaces we study in physics, we simplify the abstract mathematical notation and use Dirac’s bra-ket notation.

Dual spaces

For a vector space $V$ over a field $F$ the dual $V^\ast$ is the space of linear functionals $\phi$ on $V$:

$$ \phi : V \to F $$

A bilinear form on $V$ gives us a mapping from $V$ to $V^\ast$:

$$ v \to \alpha_v = ⟨v, \cdot⟩ $$

For $F=\cnums$ sesquilinearity and conjugation takes a natural part in this construction.

Dirac’s notation

Because of the isomorphism above one can consider the two elements in an inner product as elements from different vector spaces:

$$\alpha_\phi(v) = ⟨\phi, v⟩ $$

In Dirac’s notation, the functionals are called bras $\bra{\phi} = \alpha_\phi$ and the elements of $V$ kets $\ket{\phi} = \phi$. The inner product (conjugate linear in the first argument!) of the two is written as:

$$⟨\phi \vert v⟩ = \alpha_\phi(v) $$

Entanglement and density matrices

Fundamental differences between classical and quantum physics

Some history

Old theory

New theory

Modern theory

Questions / holes in my understanding

Path ahead

Reading