Linear Algebra and Identity Operator Generalized to 3D

[Electric to be]

I'm just getting into 3D quantum mechanics in my class, as in the hydrogen atom, particle in a box etc.

But we have already been thoroughly acquainted with 1D systems, spin-1/2, dirac notation, etc.

I am trying to understand some of the subtleties of moving to 3D. In particular, for any arbitrary state

|S(t)>, in one dimension we can use the identity operator to do: |S(t)> = ∫ dx |x><x|S(t)> or ∫ dp |p><p|S(t)> , which is basically just saying we can represent our state in terms of any of the different basis states. Perhaps momentum or position representation, or perhaps something else.

Furthermore, this kind of notation can be used to "derive" the probability for continuous and discrete operators in terms of the projection operator. For example, P = | ∫dx |x><x|S(t)> |², integrated between x = a and x = b will give the probability for finding particle between a and b.

I know how to generalize the probabilities to 3 dimensions in terms of a normal integral over x,y and z, but I just want to see how the appropriate dirac notation linear algebra looks in three dimensions, as I am having trouble deducing the identity and projection operators in 3D.

Thanks.

 

[Cryo]

Identity: $\hat{1} = \int d^3 r | \mathbb{r}\rangle\langle \mathbb{r}|$
Completeness: $\delta^{(3)}\left(\mathbb{r}-\mathbb{r'}\right)=\langle \mathbb{r} | \mathbb{r'} \rangle$

For a generic state $|S \rangle=\int d^3 r |\mathbb{r}\rangle\langle \mathbb{r}| S\rangle$

From my memory Sakurai's Modern Quantum Mechanics was quite good for this notation.