# Quantization of energy in semi-infinite well

Consider an electron with total energy $E>V_2$ in a potential with $$V(x)= \begin{cases} \infty & x< 0 \\ V_1 & 0< x< L \\ V_2 & x>L \end{cases}$$ where $V_2>V_1>0$.

We can determine that $$\phi_E(x)= \begin{cases} 0 & x< 0 \\ A\sin(kx)+B\cos(kx) & 0< x< L \\ Ce^{qx}+De^{-qx} & x>L \end{cases}$$ where $k^2=\frac{2m_e E}{\hbar}$ and $q=k\sqrt{V_1-E}$.

We can also apply the boundary condition at $x=0$ to determine that $$\phi_e(x)=A\sin(kx)$$ for $x\in[0,L]$.

We can also apply boundary conditions at $x=L$ to find that $$A\sin(kL)=De^{-qL}$$ $$Ak\cos(kL)=-Dqe^{-qL}$$ (since $C=0$ due to the corresponding positive exponent), and $$k\cot(kL)=-q$$

I'm stuck with the question: are the energy states with $E>V_2$ quantized?

I can see that, because the boundary condition at $x=L$ is not homogeneous, we cannot determine the eigenvalues in discrete form. Does this mean that the energy states are not quantized in this case?

Would appreciate some help.

• A free electron can have any energy, frequency and momentum, the same is true of massless objects such as photons. The same electron or photon in a bound or restrained environment has discrete energy and frequency levels, this arises formally because in this case the equations of Quantum Theory have to obey boundary conditions. The same consideration applies in String Theory, however a hanging rope has discrete frequencies, this may be the concept your lecturer/tutor wants you to consider. – Arif Burhan Jul 19 '16 at 5:16

## 1 Answer

$$A\sin(k_iL)=De^{-qL}$$ $$Ak_i\cos(k_iL)=-Dqe^{-qL}$$ $$k_i\cot(k_iL)=-q$$ Insert the values for $k_i$ and $q$: $$[2m(V_1-E_i)/\hbar^2]^{1/2}\cot[2m(V_1-E_i)/\hbar]^{1/2}L=-[2mE_i/\hbar^2]^{1/2}$$ The allowed energy levels ($E_i<V_2$) for bound states can be determined by numerical solution of that equation. But particles with energy $E>V_2$ can not be bound (contained in the well). Such a particle, coming in from the right e.g., would simply bounce off the infinite potential wall.