Daily Archives: Thu, 25 June 2015

Adding a constant to the potential introduces a phase factor

Required math: algebra, calculus (partial derivatives and integration by parts), complex numbers

Required physics: Schrödinger equation, probability density

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.8.

The time-independent Schrödinger equation in one dimension can be separated into two equations as follows:

\displaystyle   -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi(x)}{dx^{2}}+V(x)\psi(x) \displaystyle  = \displaystyle  E\psi(x)\ \ \ \ \ (1)
\displaystyle  i\hbar\frac{d\Xi(t)}{dt} \displaystyle  = \displaystyle  E\Xi(t) \ \ \ \ \ (2)

and the general solution is

\displaystyle  \Psi\left(x,t\right)=\psi\left(x\right)\Xi\left(t\right) \ \ \ \ \ (3)

The time component can be solved as

\displaystyle  \Xi\left(t\right)=Ce^{-iEt/\hbar} \ \ \ \ \ (4)

where {C} is the constant of integration.

If we add a constant (in both space and time) {V_{0}} to the potential, then the original Schrödinger equation becomes

\displaystyle   -\frac{\hbar^{2}}{2m}\frac{d^{2}\Psi}{dx^{2}}+V(x)\Psi+V_{0}\Psi \displaystyle  = \displaystyle  i\hbar\frac{\partial\Psi}{\partial t}\ \ \ \ \ (5)
\displaystyle  -\frac{\hbar^{2}}{2m}\frac{d^{2}\Psi}{dx^{2}}+V(x)\Psi \displaystyle  = \displaystyle  i\hbar\frac{\partial\Psi}{\partial t}-V_{0}\Psi \ \ \ \ \ (6)

Applying separation of variables gives us

\displaystyle   -\frac{\hbar^{2}}{2m}\frac{1}{\psi(x)}\frac{\partial^{2}\psi(x)}{\partial x^{2}}+V(x) \displaystyle  = \displaystyle  E\ \ \ \ \ (7)
\displaystyle  i\hbar\frac{1}{\Xi(t)}\frac{\partial\Xi}{\partial t}-V_{0} \displaystyle  = \displaystyle  E \ \ \ \ \ (8)

[Since {V_{0}} is independent of both {x} and {t}, we could put it in either the {\psi\left(x\right)} or the {\Xi\left(t\right)} equation, but putting it in the {\Xi} equation eliminates it from the more complex {\psi} equation, so we’ll do that.]

The solution to 8 is now

\displaystyle  \Xi\left(t\right)=Ce^{-i\left(E+V_{0}\right)t/\hbar} \ \ \ \ \ (9)

so we’ve introduced a phase factor {e^{-iV_{0}t/\hbar}} into the overall wave function {\Psi}. For the time-independent Schrödinger equation, all quantities of physical interest involve multiplying the complex conjugate {\Psi^*} by some operator {\hat{Q}\left(x\right)} that depends only on {x}, operating on {\Psi}. That is, we’re interested only in quantities of the form

\displaystyle   \Psi^*\left[\hat{Q}\left(x\right)\Psi\right] \displaystyle  = \displaystyle  \left|C\right|^{2}e^{+i\left(E+V_{0}\right)t/\hbar}e^{-i\left(E+V_{0}\right)t/\hbar}\psi^*\left[\hat{Q}\left(x\right)\psi\right]\ \ \ \ \ (10)
\displaystyle  \displaystyle  = \displaystyle  \left|C\right|^{2}\psi^*\left[\hat{Q}\left(x\right)\psi\right] \ \ \ \ \ (11)

Thus the phase factor disappears when calculating any physical quantity.