When Is It Appropriate To Use The Ladder Operator Method in Quantum Mechanics? I'm trying to understand when it is intuitively obvious that the ladder method would be best used to tackle a problem in quantum mechanics. 
 A: Ladder operators might be one method of a solution if


*

*the system has a discrete set of eigenvalues of an observable operator (hamiltonian, angular momentum, etc)

*you can establish a non-zero commutator relationship between the ladder operators (usually an operator and its adjoint)

*the observable operator can be constructed from the ladder operators, but generally don't commute with the observable.

*look for symmetry in the system


This is probably not an exhaustive list, but the best teacher here is experience. After you see several examples, you begin to get a sense that ladder operators might work.
It's not really "intuitively obvious" without the experience. "Obvious" to an expert = "quite elegant" to the first year grad student = "WHAT?" to the sophomore.
A: The construction of ladder operators is very closely tied to the construction of representations of Lie algebras.  Thus, whenever you have a Lie algebra lurking around, you can usually construct ladder operators.  The harmonic oscillator and angular momenta are examples of this, with  $\frak{hw}(1)$ and $\mathfrak{su}(2)$ (or $\mathfrak{so}(3)$) the corresponding Lie algebras.  
There are a great many examples where problems have underlying symmetries of the Lie type so it is quite frequent to construct such operators. One can do so for instance in the 3D harmonic oscillator, with $\mathfrak{su}(3)$ as the appropriate Lie algebra: an example of application would be the nuclear $\mathfrak{su}(3)$ model: 

Arima, A. "Elliott's SU (3) model and its developments in nuclear
  physics." Journal of Physics G: Nuclear and Particle Physics 25.4
  (1999): 581

and its extension to interacting boson models (IBMs).
Since many differential equation in physics have an underlying algebraic structure, it is very often possible to construct ladder operators.  A good example is the $\mathfrak{su}(1,1)$ structure of many radial matrix elements as in 

Armstrong Jr, Lloyd. "O (2, 1) and the harmonic oscillator radial function." Journal of Mathematical Physics 12.6 (1971): 953-957,

or

Chacón, E., D. Levi, and M. Moshinsky. "Lie algebras in the Schrödinger picture and radial matrix elements." Journal of Mathematical Physics 17.10 (1976): 1919-1929.

There's a pretty sweet generalization, reviewed in 

Infeld, Leopold, and T. E. Hull. "The factorization method." Reviews of modern Physics 23.1 (1951): 21

which allows for the construction of creation- and destruction-like operators when the Hamiltonian is factorizable.  This type of factorization is the basis of supersymmetric quantum mechanics:

Cooper, Fred, Avinash Khare, and Uday Sukhatme. "Supersymmetry and quantum mechanics." Physics Reports 251.5-6 (1995): 267-385.

A: Two excellent examples of the use of ladder operators can be found in Introduction to Quantum Mechanics (3rd Ed, Griffiths): 
1. The 1D harmonic oscillator, in chapter 2 (pg 43) and
2. The spherical harmonics for the total angular momentum operator (pg 159).
In both of these, there is a common theme. If there exists two operators such that one takes an arbitrary eigenfunction (of the operator we are trying to find the eigen configuration for) to the next one and the other takes the eigenfunction to the previous one, then it is possible to use ladder operators. In both cases, Griffiths just gives operator(s) that work as the ladders and there isn’t a “god given” method to find that they are supposed to be, so it can be challenging to find the appropriate ladder operators for a problem. 
