One-particle, time-dependent Schrodinger equation

The one-particle, time-dependent Schrodinger equation is a partial differential equation whose solutions are the one-particle, time-dependent wave functions of quantum-mechanical systems.

Even though the equation is widely regarded as a postulate, we can derive it using a general travelling wave equation \psi(x,t)=Acos\frac{2\pi}{\lambda}(x-ct). Since cosine is an even function, Acos\frac{2\pi}{\lambda}(x-ct)=Acos\frac{2\pi}{\lambda}(ct-x), which in the complex square-integrable form is: \psi(x,t)=Ae^{-i\frac{2\pi}{\lambda}(ct-x)} . Since c=v\lambda, we have \psi(x,t)=Ae^{-i2\pi( vt-\frac{x}{\lambda} )}. Substituting Planck’s relation and de Broglie’s hypothesis in the wave equation, which is a mathematical description of the properties of a quantum-mechanical particle, we have \psi(x,t)=Ae^{-\frac{i}{\hbar}(Et-xp )}, where \hbar=\frac{h}{2\pi}.

The total energy of the particle is: E=T+V=\frac{p^{2}}{2m}+V, and so

E\psi=\frac{p^{2}\psi}{2m}+V\psi\; \; \; \; \; \; \; \; 54

To develop an expression for E\psi, we find the partial derivative of \psi with respect to t:

\frac{\partial}{\partial t}Ae^{-\frac{i}{\hbar}(Et-xp)}=Ae^{\frac{i}{\hbar}xp}\frac{\partial}{\partial t}e^{-\frac{i}{\hbar}Et}=-A\frac{iE}{\hbar}e^{-\frac{i}{\hbar}(Et-xp)}=-\frac{iE}{\hbar}\psi

E\psi=i\hbar\frac{\partial\psi}{\partial t}\; \; \; \; \; \; \; \; 55

As for p^{2}\psi, we find the the 2nd-order partial derivative of \psi with respect to x:

\frac{\partial^{2}}{\partial x^{2}}Ae^{-\frac{i}{\hbar}(Et-xp)}=Ae^{-\frac{i}{\hbar}Et}\frac{\partial^{2}}{\partial x^{2}}e^{\frac{i}{\hbar}xp}=\left ( \frac{i}{\hbar}p\right )^{2}Ae^{-\frac{i}{\hbar}(Et-xp)}=-\frac{p^{2}}{\hbar^{2}}\psi

p^{2}\psi=-\hbar^{2}\frac{\partial^{2}\psi}{\partial x^{2}}\; \; \; \; \; \; \; \; 56

Substituting eq55 and eq56 in eq54, we have

i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V\psi\; \; \; \; \; \; \; \; 57

Eq57 is the one-particle, one-dimensional, time-dependent Schrodinger equation, which has the general solution \psi(x,t)=\psi(x)e^{-\frac{i}{\hbar}Et}.



Show that \psi(x,t)=\psi(x)e^{-\frac{i}{\hbar}Et} is a solution to eq57.


For LHS of eq57

i\hbar\frac{\partial\psi(x,t)}{\partial t}=-i\hbar\psi(x)i\frac{E}{\hbar}e^{-i\frac{Et}{\hbar}}=E\psi(x,t)


\left [ -\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+V\right ]\psi(x,t)=E\psi(x,t)




Next article: classical angular momentum
Previous article: 1-D classical wave equation
Content page of quantum mechanics
Content page of advanced chemistry
Main content page

Leave a Reply

Your email address will not be published. Required fields are marked *