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Difference Equation vs. Differential Equation

A new way to time evolve

We’ve so far looked at the time evolution of a quantum state by separating out the function in terms of position and time, using a differential approach, where we multiply the x positions of a wavevector (in position space) by the time-dependence:

$$\begin{equation}\label{tdse} \hat{H}\Psi(x,t) = E\Psi(x,t) \end{equation}$$

Where the time dependence is represented as: ${T_n(t) = e^(\frac{−i(E_n)t}{\hbar}}$

Instead of using a differential equation to solve the Hamiltonian and find an energy eigenvalue using this formula: $ \hat{H} = \frac{-hbar^2}{2m}\frac{\partial^2}{\partial x^2} + {V(x)} $, we can instead define a difference equation to better understand the time evolution of a quantum state.

Where we have previously discritized space using some arbitrary quantity {dt}, we can now compare the wavefunctions at two different times in order to better understand time evolution. $\begin{equation}\label{woo} \hat{H}\Psi(x,t) = i{\hbar}\frac{\partial}{\partial t}\Psi(x,t) \end{equation}$

This expression can now be expressed in the difference form by comparing the wavefunction {\psi} at two different times: $\begin{equation}\label{wowwow} \hat{H}\Psi(x,t) = i{\hbar}\frac{\Delta \Psi(x,t)}{\Delta t}\end{equation}$

With algebraic manipulation of this expression, we can ultimately solve for what the wavefunction appears as at a later time. Because ${-i}$ and ${\hbar}$ are defined as constants, we can rearrange the equation.

$$\begin{equation}\label{another} \hat{H}\Psi(x,t) = i{hbar} \frac{\Psi(x, t + \Delta t) - \Psi(x,t)}{\Delta t} \end{equation}$$

$ \Psi(x, t + \Delta t) = \frac{\Delta t \hat{H} \Psi(x, t)}{i \hbar} + \Psi(x, t + \Delta t) $

The function which is {\psi} at the initial time can then be factored out of this expression above such that:

$ \Psi(x, t + \Delta t) = (\frac{\Delta t \hat{H}}{i \hbar} + 1) \Psi(x, t) $

This form allows for a new translation into the linear algebra that we’ve been using so far as well. The first half of the expression can simplify to (${i \hbar}$ multiplied by the product of the Hamiltonian matrix which takes the second derivative of a different matrix and the identity matrix of the same dimensions as the Hamiltonian) all then multiplied by the original wavevector. These mathematical manipulations would then yield a new wavevector of the original wavevector at a different time - $ {\Psi (x, t + \Delta t)} $.