# How can one find the energy eigenfunctions of a particle in a finite square well via the Klein-Gordon equation?

It is said that Klein-Gordon equation is a relativistic version of the Schrodinger equation. In Schrodinger equation, it is straightforward to include potential energy. But for K-G eqn things seem to be more complicated.

To put it specifically, how can one find the energy eigenfunctions of a particle in a finite square well via Klein-Gordon equation?

• In terms of the logical role it plays in QFT it is not true that the K-G equation is a relativistic version of the Schrodinger equation and the sooner you get this idea our of your head the better off you will be when you learn QFT. – pho Dec 20 '10 at 14:34
• BUt in terms of relativistic particle mechanics in which the number of particles is fixed, it is true to think of it as the relativistic version of the Schrodinger equation. But thinking of it that way introduced problems, such as negative probabilities, which Weisskopf and Pauli got rid of when they re-interpreted the K-G equation as not a quantum eq., but a classical one, which they then quantised to get a quantum field, which introduces all the usual problems of the infinities of Quantum Field Theory etc. – joseph f. johnson Dec 20 '11 at 8:32
• For a connection between Schr. eq. and Klein-Gordon eq, see e.g. A. Zee, QFT in a Nutshell, Chap. III.5, and this Phys.SE post plus links therein. – Qmechanic Dec 15 '12 at 18:23

K-G is a relativistic equation. So if you want to couple it to external fields you have to use a vector potential $A_\mu$ which will modify the partial-derivative, so that the Klein-Gordon equation:

$$\left( \partial^\mu \partial_\mu + m^2 \right)\phi = 0$$

becomes:

$$\left ( D^\mu D_\mu + m^2 \right) \phi = 0$$

where $D_\mu = \partial_\mu + \imath A_\mu$.

Then it remains to figure out what $A_\mu$ will correspond to a given potential - finite square well or otherwise. For this you will likely have to fix a gauge before you can identify the potential as such.

A square well potential will correspond to $A_\mu = (V(x), 0,0,0)$, where $V(x) = V_0 \,\forall |x| > \pm a$ and $V(x) = 0, x \in [-a,a]$. The resulting behavior is a little counter-intuitive, yielding something known as the Klein paradox.

For further details see ch. 4 for this excellent thesis on this problem by D. Schaffer.

• Just a remark (nothing wrong with your answer): you don't need a vector potential. This is only one possible theory (electromagnetic actually). You could also insert a term such as $\bar \Psi \phi \Psi$ (Yukawa interaction) coupling to fermions and lots of others (actually the only limitation is that the theory be Lorentz-invariant on this level). – Marek Dec 20 '10 at 8:01
Now, as to your specific eigenvalue-problem query. Pauli assigns that exact problem in his lectures on Wave Mechanics. The method of solution is the same as for the Scrhoedinger eq. Solve the free equ. in the half-infinite region on the left of the potential well. Solve it on the right. In the well, since the potential is constant, the same method (plug in an exponential function and solve for the parameters) works. Now match boundary conditions at the two interfaces, and see which values of the parameters allow exponential decay in the infinite regions so that the wave function is $L^2$. Once you choose a parameter in the left region, the parameters in the well and the right region are forced on you by matching the boundary conditions at the interfaces.