Time Propagation: Evaluating "exp(-i*H*t)

[Incognition]

I'm running a simple limiting case of a simulation with the expression:

exp(-i*H*t), where H is the hamiltonian, in this case a 2x2 matrix of zeros. Should this evaluate to 0, or to [ 1 1 ; 1 1] ?

Right now my sim uses the latter, which I'm sure is wrong. What do you think? Thanks in advance.

 

[CompuChip]

By definition
$$\exp(A) = 1 + A + \frac{1}{2} A^2 + \frac{1}{6} A^3 + \cdots$$

So exp(0) = 1
with 1 the identity matrix $$\begin{pmatrix}1&0\\0&1\end{pmatrix}$$.

 

[denian]

Based on the information provided, it appears that the simulation is using a 2x2 matrix with all zeros as the Hamiltonian. In this case, the evaluation of exp(-i*H*t) should result in a 2x2 matrix of all ones, not [1 1; 1 1]. This is because the exponential of a zero matrix is the identity matrix, which in this case would be [1 0; 0 1]. Therefore, the correct evaluation should be [1 0; 0 1].

Additionally, it is important to note that the time propagation of a system is a fundamental concept in quantum mechanics and is crucial in understanding the behavior of quantum systems. The expression exp(-i*H*t) represents the time evolution operator, which describes how a quantum state changes over time. It is derived from the Schrödinger equation and is a fundamental tool in simulating and understanding quantum systems.

In summary, the correct evaluation for exp(-i*H*t) in this case should be [1 0; 0 1], and it is important to use the correct expression in simulations to accurately model quantum systems. I would recommend double checking the implementation of the simulation and ensuring that the correct mathematical expressions are being used to accurately represent the system being studied.