Schrodinger Equation | Quantum Mechanics

Consider a particle of mass ‘m’ is moving along the x axis, under the potential field:

$V\left(x\right)$

which is independent of time. The total energy of the system can be written as:

$E=E_{k}+E_{p}=\frac{P(x)^{2}}{2m}+V\left(x\right)$

Now, the 1-D time-dependent Schrodinger equation for the system can be written as:

$\hat{H}\psi=\hat{E}\psi$

$\Longrightarrow-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V(x)\psi=i\hbar\frac{\partial\psi}{\partial\text{${t}$}}$

Solution of 1-D time-dependent Schrodinger equation:

Let,

$\psi(x,t)=X(x)T(t)$

Replacing this in the 1-D time-dependent Schrodinger equation:

$\Longrightarrow\frac{-\hbar^{2}}{2m}\text{${T}$}\frac{\partial^{2}\psi}{\partial x^{2}}+VXT=i\hbar X\frac{\partial T}{\partial\text{t}}$

$\Longrightarrow-\frac{\hbar^{2}}{2m}\frac{1}{X}\frac{d^{2}X}{dx}+V=\frac{i\hbar}{T}\frac{\partial T}{\partial\text{t}}=E(constant)$

Time-dependent part:

$i\hbar\frac{1}{T}\text{${\tau}$}\frac{\partial T}{\partial t}=E$

$\Longrightarrow\frac{\partial T}{T}=\frac{E}{i\hbar}\partial\text{t}=-\frac{i}{\hbar}Edt$

$\Longrightarrow T(t)=Ce^{-t\frac{E}{\hbar}t}$

Therefore, the time-dependent part of the solution of the 1-D Schrodinger equation is independent of:

$V\left(x\right)$.

Time-independent part:

$-\frac{\hbar^{2}}{2m}\frac{1}{X}\frac{d^{2}X}{dx}+V=E$

$\Longrightarrow-\frac{\hbar}{2m}\frac{d^{2}X}{dx^{2}}+VX=EX$

$\Longrightarrow\frac{d^{2}X}{dx^{2}}+\frac{2m}{\hbar^{2}}\frac{d^{2}X}{dx^{2}}\left[E-V(x)\right]X=0$

Therefore, the time-independent part of the solution of the 1-D Schrodinger equation depends on the nature of: $V\left(x\right)$.

Time-Dependent

The solution of the 1-D time-dependent Schrodinger equation can be written as:

$\psi(x,t)=X(x)T(t)=X(x)e^{-iEt/\hbar}$

Now, if:

$\psi_{1}(x,t),\psi_{2}(x,t)…..\psi_{n}(x,t)$

are the solutions of the 1-D Schrodinger equation corresponding to energies

$E_{1}E_{2}E_{3}…..E_{N},$

then the general solution is:

$\psi(x,t)=\psi_{1}(x,t)+\psi_{2}(x,t)+….$

$=C_{1}X_{1}(x)e^{-iE_{1}\frac{t}{\hbar}}+C_{2}X_{2}(x)e^{-iE_{2}\frac{t}{\hbar}}+C_{3}X_{3}(x)e^{-iE_{3}\frac{t}{\hbar}}+….$

$=\sum_{n}C_{n}X_{n}(x)e^{-iE_{2}\frac{t}{\hbar}}$