# Quantum Mechanics Intermediates¶

## Tensor Product Space¶

This part has been moved to Tensor Product Space

## Density Matrix¶

The density matrix operator is

The state \(\ket{\psi(t)}\) can be projected onto a set of basis \(\ket{\phi_n}\),

Using the basis \(\ket{\phi_n}\), the density matrix is represented as \(\rho_{mn}\)

For pure states, density matrix and kets carries the same information.

Density matrix can be used to denote mixed states easily. For mixed states, we write down the density matrix using a mixture of pure states \(\ket{\psi_i(t)}\)

Mixed states are like mixture models in statistics.

## Angular Momentum¶

### Angular Momentum¶

For an new operator, we would like to know

- Commutation relation: with their own components, with other operators;
- Eigenvalues and their properties;
- Eigenstates and their properties;
- Expectation and classical limit.

#### Definition of Angular Momentum¶

In classical mechanics, angular momentum is defined as

One way of defining operator is to change position and momentum into operators and check if the operator is working properly in QM. So we just define

It is Hermitian. So it can be an operator. We also find

**More generally, we can define angular momentum as**

We can prove that

So they can have the same eigenstates

To find the constraints on these eigenvalues, we can use positive definite condition of certain inner porducts, such as,

where

and we have

It’s easy to find out that

i.e., \(\hat J_{\pm}\ket{\lambda m}\) is eigenstate of \(\hat J_z\).

Follow the plan of finding out the bounds through these positive inner products, we can prove that

#### Eigenstates of Angular Momentum¶

As we have proposed, the eigenstates of both \(\hat J_z\) and \(\hat{\vec J}^2\) are \(\ket{j,m}\), where \(j=0,1,2,\cdots\) and \(m=-j,-j+1,\cdots, j-1,j\).

We can also find out the wave function in \({\ket{\theta,\phi } }\) basis. Before we do that, the definition of this basis should be made clear. This basis spans the surface of a 3D sphere in Euclidean space and satisfies the following orthonormal and complete condition.

Now we have an arbitary state \(\ket{\psi}\),

Then we define

which is the spherical harmonic function.

Then

So as long as we find out what \(\psi _ {lm}\) is, any problem is done.