Schrödinger Equation in Position Space

From bio-physics-wiki

Jump to: navigation, search

The abstract Schrödinger Equation as we introduced it as a postulate is in three dimensions

\begin{align} i \, \hbar \frac{\partial}{\partial t} |\psi(t) \rangle & = \left( \frac{\hat{\mathbf{p}}^2}{2m} + V(\hat{\mathbf{r}}) \right) |\psi(t) \rangle \end{align} Representing $|\psi(t) \rangle$ in the position basis by multiplying on the left with the bra $\langle \mathbf{r} |$ we get \begin{align} i \, \hbar \frac{\partial}{\partial t} \langle \mathbf{r} |\psi(t) \rangle & = \langle \mathbf{r} | \left( \frac{\hat{\mathbf{p}}^2}{2m} + V(\hat{\mathbf{r}}) \right) |\psi(t) \rangle \end{align} At the left hand side $\langle \mathbf{r} |\psi(t) \rangle$ is just $\psi (\mathbf{r},t)$. $\hat{\mathbf{p}}^2$ the momentum operator in the position basis is just $(i \hbar \nabla )^2=- \hbar^2 \nabla ^2$. $\langle \mathbf{r} | V(\hat{\mathbf{r}}) |\psi(t) \rangle$ with the operator $\mathbf{r}$ acting on the bra $\langle \mathbf{r} |$ gives the eigenvalue $\mathbf{r}$, thus $\langle \mathbf{r} | V(\hat{\mathbf{r}}) |\psi(t) \rangle=\langle \mathbf{r} | V(\mathbf{r}) |\psi(t) \rangle=V(\mathbf{r}) \langle \mathbf{r} | \psi(t) \rangle$. Now we can rewrite the Schrödinger Equation as \begin{align} i \, \hbar \frac{\partial}{\partial t} \langle \mathbf{r} |\psi(t) \rangle & = \left( \frac{-\hbar^2 }{2m} \nabla ^2+ V(\mathbf{r}) \right) \langle \mathbf{r}|\psi(t) \rangle\\ \end{align}

In the position basis the Schrödinger Equation becomes \begin{align} i \, \hbar \frac{\partial}{\partial t} \psi(\mathbf{r},t)& = \left( \frac{-\hbar^2 }{2m} \nabla ^2+ V(\mathbf{r}) \right) \psi(\mathbf{r},t)\\ \end{align}

We solve the Equation by Separation of Variables with the ansatz $\psi(\mathbf{r},t)=\Psi(\mathbf{r})f(t)$. \begin{align} i \, \hbar \frac{\partial}{\partial t} \Psi(\mathbf{r})f(t)& = \left( \frac{-\hbar^2 }{2m} \nabla ^2+ V(\mathbf{r}) \right) \Psi(\mathbf{r})f(t)\\ i \, \hbar \Psi(\mathbf{r})f'(t)& = \frac{-\hbar^2 }{2m} \nabla ^2 \Psi(\mathbf{r})f(t) + V(\mathbf{r}) \Psi(\mathbf{r})f(t) \\ \end{align} dividing by $\Psi(\mathbf{r})f(t)$ we have \begin{align} i \, \hbar \frac{\partial}{\partial t} \Psi(\mathbf{r})f(t)& = \left( \frac{-\hbar^2 }{2m} \nabla ^2+ V(\mathbf{r}) \right) \Psi(\mathbf{r})f(t)\\ i \, \hbar \frac{f'(t)}{f(t)} & = \frac{-\hbar^2 }{2m} \frac{\nabla ^2 \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + V(\mathbf{r}) \\ \end{align} which we can set equal to a constant. \begin{align} i \, \hbar \frac{f'(t)}{f(t)} =E \end{align} \begin{align} \frac{-\hbar^2 }{2m} \frac{\nabla ^2 \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + V(\mathbf{r}) = E \\ \end{align}

The first equation is an ODE and easyly solved \begin{align} f(t)= C \cdot e^{- i E\cdot t /\hbar}= C \cdot e^{-i \omega t} \end{align} where we used $E=h \nu=\hbar \omega$. $f(t)$ is in fact the time development operator we introduced with the postulates

$f(t)$ is the unitary time development operator \begin{align} f(t)= U(t,t_0)= e^{- i E\cdot (t-t_0) /\hbar}= e^{-i \omega (t-t_0)} \end{align} which is just a phase factor.

The other differential equation is called

the Stationary Schrödinger Equation \begin{align} \frac{-\hbar^2 }{2m} \nabla ^2 \phi_n(\mathbf{r}) + V(\mathbf{r})\phi_n(\mathbf{r}) = E_n \phi_n(\mathbf{r})\\ \end{align} or \begin{align} \left( \frac{-\hbar^2 }{2m} \nabla ^2 + V(\mathbf{r}) \right) \phi_n(\mathbf{r}) = E_n \phi_n(\mathbf{r}) \\ \end{align} which is an eigenvalue equation with the eigenvalue $E_n$.

The general solution of the Stationary Schrödinger Equation is given by the superpositon of the $\phi_n$'s

\begin{align} \Psi(\mathbf{r},t_0)= \sum_ n c_n \phi_n(\mathbf{r},t_0)\\ \end{align} $\Psi(\mathbf{r},t_0)$ must be spcified with the initial conditions. The sum can also become an integral if the basis is continuous. When are the Eigenvalues $E_n$ (energy levels) discrete, when they are continuous?

A free particle that is not constrainted by any potential or other barriers has a continuous set of Eigenvalues $E$. Any confinement of the particle quantises the energy levels to a discrete set of Eigenvalues $E_n$.

The solution to the general Schrödinger Equation is

\begin{align} \Psi(\mathbf{r})f(t)= \sum_ n c_n \, e^{- i E_n\cdot (t-t_0) /\hbar} \, \phi_n(\mathbf{r},t_0)\\ \end{align}

Notice, that the sum \begin{align} \Psi(\mathbf{r})f(t)= \sum_ n c_n \, e^{- i \omega_n \cdot (t-t_0)} \, \phi_n(\mathbf{r},t_0)\\ \end{align} contains a superposition of phase factors with different angular frequencies $\omega_n$. This reflects the phenomenon of quantum interference.

Video Lectures:

  • V. Balakrishnan - Quantum Physics Lec 8 [1]

Further Reading:

  • D. Griffiths - Introduction to Quantum Mechanics
  • Yoav Peleg et al. - Quantum Mechanics (Schaums Outline: with many solved exercises)
  • Claude Cohen-Tannoudji - Quantenmechanik Band 1.