# Quantum Mechanics Intermediates¶

## Tensor Product Space¶

This part has been moved to Tensor Product Space

## Angular Momentum¶

### Angular Momentum¶

For an new operator, we would like to know

1. Commutation relation: with their own components, with other operators;

2. Eigenvalues and their properties;

3. Eigenstates and their properties;

4. Expectation and classical limit.

#### Definition of Angular Momentum¶

In classical mechanics, angular momentum is defined as

$\vec L = \vec X \times \vec P .$

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

$\hat {\vec L} = \hat {\vec X}\times \hat{\vec P}.$

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

$\hat{\vec L}\times \hat{\vec L} = i \hbar \hat{\vec L}$
$\left[\hat L_i,\hat L_j\right] = \sum_k i\epsilon_{ijk}\hat L_k .$

More generally, we can define angular momentum as

$\left[\hat J_i, \hat J_j\right] = i\hbar \sum_k \epsilon_{ijk} \hat J_k$

We can prove that

$\left[ \hat J^2,\hat J_z \right] = 0.$

So they can have the same eigenstates

$\hat J_z \ket{\lambda m} = m\hbar \ket{\lambda m}$
$\hat J^2 \ket{\lambda m} = \lambda^2 \hbar^2 \ket{\lambda m}$

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

$\bra{\psi} \hat J_+ \hat J_- \ket{\psi} \geq 0$
$\bra{\psi} \hat J_- \hat J_+ \ket{\psi} \geq 0$

where

$\hat J_{\pm} = \hat J_x \pm i \hat J_y$

and we have

$\left[\hat J_+, \hat J_-\right] = 2 \hbar \hat J_z$
$\left[\hat J_z, \hat J_{\pm} \right] = \pm \hbar \hat J_{\pm}.$

It’s easy to find out that

$\hat J_z (\hat J_{\pm}\ket{\lambda m}) = (m\pm 1) \hbar (\hat J_{\pm} \ket{\lambda m})$

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

$\hat J^2\ket{jm} = j(j+1)\hbar^2 \ket{jm}$
$\hat J_{\pm}\ket{jm} = \sqrt{j(j+1)-m(m\pm 1)} \hbar \ket{j,m\pm 1}$

#### 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.

$\int \mathrm d \Omega \braket{\theta',\phi'}{\theta,\phi} = \delta(\cos\theta'-\cos\theta,\phi'-\phi) \int \mathrm d \Omega \ket{\theta',\phi'}\bra{\theta,\phi} = 1$

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

$\begin{split}\ket{\psi} &= \sum _ {l,m} \psi _ {lm}\ket{l,m} \\ &= \sum _ {l,m} \int \mathrm d \Omega \ket{\theta',\phi'}\bra{\theta,\phi} \psi _ {lm}\ket{l,m} \\ &= \sum _ {l,m} \int \mathrm d \Omega \ket{\theta',\phi'} (\braket{\theta,\phi}{l,m} ) \psi _ {lm} \\\end{split}$

Then we define

$\braket{\theta,\phi}{l,m}=Y_l^m(\theta,\phi)$

which is the spherical harmonic function.

Then

$\begin{split}\ket{\psi} &= \sum _ {l,m} \psi _ {lm} \int \mathrm d \Omega Y_l^m(\theta,\phi) \ket{\theta',\phi'} \\\end{split}$

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