For a particle of mass $m$ moving in a potential $V(x,t)$ which can be time-dependent, the time-dependent Schrödinger equation in one dimension (1D TDSE) is:

$$ \frac{- \hbar^2}{2 m} \frac{\partial^2 \Psi}{\partial x^2} + V(x,t) \Psi = i \hbar \frac{\partial \Psi}{\partial t} $$

• The solution is $\Psi(x,t)$ the wavefunction of a particle
• If $V(x)$ is a function of only $x$ then the equation can be solved by separation of variables

Solving by separation of variables

we assume $\Psi (x,t) = \Psi (x) T (t)$ and sub in

$$ \begin{aligned}\left ( \frac{-\hbar^2}{2 m} \frac{d^2 \Psi}{d x^2} + V(x) \Psi(x) \right ) T(t) &= i \hbar \Psi (x) \frac{dT(t)}{dt}\\\mathrm{i.e.} \qquad \qquad \quad \underbrace{\frac{1}{\Psi(x)} \frac{- \hbar^2}{2 m} \frac{d^2 \Psi}{dx^2} + V(x)}{\text{depends on $x$ only}} &= \underbrace{\frac{i \hbar}{T(t)} \frac{dT(t)}{dt}}{\text{depends on $t$ only}}\end{aligned} $$

• This equation can only be satisfied for all $t$ and $x$ if both sized equal the same constant $E$ thus we get 2 equations:

1. Time equation: $\frac{i \hbar}{T(t)} \frac{d T(t)}{dt} = E$

   The solution is $T= T_0 e^{\frac{-iEt}{\hbar}} = T_0 e^{-i \omega t}$ where $\omega = \frac{E}{\hbar}$

2. Space equation: $\frac{- \hbar^2}{2 m} \frac{d^2 \Psi}{dx^2} + V(x) \Psi = E \Psi$

   The solutions $\Psi(x)$ are called stationary states

Interpretation of the wavefunction

Born suggested that the probability of finding a particle at a position between $x$ and $x+\text dx$ at time $t$ where $P(x,t)$ is the probability density is

$$ |\Psi(x,t)|^2 dx = P(x,t) dx $$

• If this interpretation is correct, because the particle must be somewhere we have:

$$ \int_{-\infty}^{+\infty} |\Psi(x,t)|^2 dx = 1 $$

Interpretation restrictions

• $\Psi$ must be finite everywhere
• $\Psi$ must be single-values ( i.e. there is only one probability at finding the particle at a given position )
• $\Psi$ and $\frac{\partial \Psi}{\partial x}$ must be continuous ( because Schrödinger's equation requires $\frac{\partial^2 \psi}{\partial x^2}$ )

These conditions mean that only certain solutions are allowed

Infinite potential well

Consider the following one dimensional finite potential well

$$ V(x) = \left \{ \begin{array}{ll} \infty &\quad \mathrm{if} \quad x<0 \\ 0 & \quad \mathrm{if} \quad 0\le x \le a\\ \infty & \quad \mathrm{if} \quad x>a \end{array} \right . $$