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Suppose I have a particle with mass $m$ and it's under potential of a certain $V(x)$. (NOT an infinite or finite potential well)

Also given is the wave function at time $t=0$, $\psi(x,0)$.

What is the method of getting from here to find the complete $\psi(x,t)$?

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There is no unique "the method" for solving this. You're basically solving the initial-value problem \begin{align} i\hbar \frac{\partial }{{\partial t}}\psi(x,t) & = \left( {\frac{{ - {\hbar ^2}}}{{2m}}\frac{{{\partial ^2}}}{{\partial {x^2}}} + V\left( x \right)} \right)\psi \left( {x,t} \right) , \\ \psi(x,0) & = \varphi(x), \end{align} and like any problem in partial differential equations, in the generic case, it is hard.

There is a wide array of relevant methods, of course:

  • you could solve it numerically,
  • you could try to find the Green's function for the problem,
  • you could use a spectral method $-$ i.e. you could solve the time-independent version of the Schrödinger equation, and then express your initial condition as a sum of those solution,

among others, and they will all have their advantages and disadvantages (starting with the fact that they will entail different interpretations of what it means to "solve" the problem), but there is no single choice that will be satisfactory for every situation.

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If you are in coordinate representation, this is the equation you need to solve with the proper initial conditions: $$i\hbar \frac{\partial }{{\partial t}}\psi \left( {x,t} \right) = \left( {\frac{{ - {\hbar ^2}}}{{2m}}\frac{{{\partial ^2}}}{{\partial {x^2}}} + V\left( x \right)} \right)\psi \left( {x,t} \right).$$

Without knowing ${V\left( x \right)}$ and $\psi \left( {x,0} \right)$ it is hard to step further. Depending on you aims, you can consider the following possibilities.

Method 1

Knowing $\psi \left( {x,0} \right)$ lets you to solve this question numerically: you substitute $\psi \left( {x,0} \right)$ into the equation $$\Delta \psi \left( {x,0} \right) = \frac{1}{{i\hbar }}\left( {\frac{{ - {\hbar ^2}}}{{2m}}\frac{{{\partial ^2}}}{{\partial {x^2}}} + V\left( x \right)} \right)\psi \left( {x,0} \right)\Delta t$$ telling you how much you should change the value of $\psi \left( {x,0} \right)$ from time $0$ to time $\Delta t$, because $$\Delta \psi \left( {x,0} \right) = \psi \left( {x,\Delta t} \right) - \psi \left( {x,0} \right).$$ Then you calculate it for every point, getting $\psi \left( {x,\Delta t} \right)$. Then you calculate again $$\Delta \psi \left( {x,\Delta t} \right) = \frac{1}{{i\hbar }}\left( {\frac{{ - {\hbar ^2}}}{{2m}}\frac{{{\partial ^2}}}{{\partial {x^2}}} + V\left( x \right)} \right)\psi \left( {x,\Delta t} \right)\Delta t,$$ and so one. This method is called the Euler method, a a numerical method for ordinary differential equations. Depending on the problem, you may want to use other methods, like the Runge-Kutta method.

Method 2

In a hopefully gracious case the given $\psi \left( {x,0} \right)$ is an energy-eigenstate. What does it mean? Try to find the solution in the form: $$\psi \left( {x,t} \right) = \phi \left( x \right) \cdot {e^{ - \frac{i}{h}Et}}.$$ This can be a solution only if $$\label{eq:energy} \tag{2} \frac{{{\hbar ^2}}}{{2m}}\frac{{{d^2}x}}{{d{x^2}}}\phi \left( x \right) + V\left( x \right) \cdot \phi \left( x \right) = E \cdot \phi \left( x \right).$$ Is that true? Calculate the left hand side first and check if it is proportional to the right hand side. If they are, you have got the exact time evolution of ${\psi}$.

Method 3

If method 2 fails you need to decompose your function into energy-eigenstates. $$\psi \left( {x,t} \right) = {c_1} \cdot \underbrace {{\phi _1}\left( x \right) \cdot {e^{ - \frac{i}{\hbar }{E_1}t}}}_{{\psi _1}\left( {x,t} \right)} + {c_2} \cdot \underbrace {{\phi _2}\left( x \right) \cdot {e^{ - \frac{i}{\hbar }{E_2}t}}}_{{\psi _2}\left( {x,t} \right)} + \ldots $$ Here, each function ${\psi_i}$ is an energy eigenstate with energy $E_i$. ${\psi_i}$ satisfies the Scrödinger equation so does their weighted sum. You know the time evolution of each ${\psi_i}$, therefore, you know the time evolution of ${\psi}$ too. To find what kind of ${\psi_i}$ and ${E_i}$ you need to use, you must solve \ref{eq:energy} for $\phi$.

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