Why use Crank-Nicolson over Matrix Exponential when solving Schrödinger's equation? For Schrödinger's equation,
$$\psi(x,t+\Delta t)=e^{-i H\Delta t}\psi(x,t)\approx\frac{1-\frac{1}{2}i H\Delta t}{1+\frac{1}{2}i H\Delta t}\psi(x,t).$$
The right-most expression is the Crank-Nicolson scheme for solving the system. However, it is only an approximation of doing matrix exponentiation.
A lot of literature (or course notes online) that I find seem to use the Crank-Nicolson scheme instead of computing $e^{-i H\Delta t}$. Is there some benefit to using the approximation given by the Crank-Nicolson scheme over doing a matrix exponentiation?
It seems to me that even if numerical algorithms to compute matrix exponential (such as scipy.linalg.expm ) use a higher-order Padé approximation - this would still be preferable to the Crank-Nicolson scheme.
 A: It is hard to compute matrix exponential. And the Crank-Nicolson approximation of evolution operator has at least two advantages besides simple enough computation. Asymptotic equalities
$$
1-iH\Delta t = e^{-iH\Delta t} + O\left( \Delta t^2\right),
$$
$$
\frac{1-\frac12iH\Delta t}{1+\frac12iH\Delta t} = 1 - iH\Delta t - \frac{\Delta t^2}2 H^2 + O\left( \Delta t^3\right) = e^{-iH\Delta t} + O\left( \Delta t^3\right)
$$
demonstrate the first one. The second advantage is strict unitarity
$$
\left(\frac{1-\frac12iH\Delta t}{1+\frac12iH\Delta t}\right)^\dagger = \frac{1+\frac12iH\Delta t}{1-\frac12iH\Delta t} = \left(\frac{1-\frac12iH\Delta t}{1+\frac12iH\Delta t}\right)^{-1},
$$
in opposition to approximate one
$$
(1- iH\Delta t)^\dagger = 1 + iH\Delta t = (1 - iH\Delta t)^{-1} + O(\Delta t^2) \neq (1 - iH\Delta t)^{-1}.
$$
Addition. I am not a computational physics specialist, so I can only suppose what is the actual reason why the Crank-Nicolson approximation is used. Not always you need or can have matrix written in computer memory. Sometimes you only have a procedure that computes matrix action on vectors. In such cases, computations of higher degrees of $H$ involved in the Pade approximation might be undesirable.
