zenterix
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- Homework Statement
- I am following the calculations in solving Schrödinger's time-independent equation for a single particle in a box.
- Relevant Equations
- $$-\frac{\hbar^2}{2m}\frac{\partial^2\psi(x)}{\partial x^2}=\psi(x)(E-V(x))\tag{1}$$
I have a question about one of the steps in the calculations.
For context about where in the calculations I am, we assume that a particle is subjected to a potential energy function that is infinite everywhere along the ##x##-axis except for a line segment of length ##l## where the potential energy is zero.
Thus, the ##x##-axis is divided into three regions where we will solve Schrödinger's equation
It can be shown that in regions I and III, the solution to (1) is just ##\psi(x)=0##.
Consider region II.
$$-\frac{\hbar^2}{2m}\frac{\partial^2\psi_{II}(x)}{\partial x^2}=\psi_{II}(x)E$$
$$\frac{\partial^2\psi_{II}(x)}{\partial x^2}=-\frac{2m}{\hbar^2}E\psi_{II}(x)$$
$$\frac{\partial^2\psi_{II}(x)}{\partial x^2}+\frac{2mE}{\hbar^2}\psi_{II}(x)=0$$
$$\frac{\partial^2\psi_{II}(x)}{\partial x^2}+k^2\psi_{II}(x)=0$$
This is a 2nd-order, linear, homogeneous ODE with
$$k^2=\frac{2mE}{\hbar^2}$$
A general (complex) solution is
$$\psi_{II}=c_1e^{x\frac{i\sqrt{2mE}}{\hbar}}+c_2e^{\frac{-ix\sqrt{2mE}}{\hbar}}$$
$$=c_1e^{ikx}+c_2e^{-ikx}$$
$$=c_1(\cos(kx)+i\sin{kx})+c_2(\cos{kx}-i\sin{kx})$$
$$=(c_1+c_2)\cos{kx}+i(c_1-c_2)\sin{kx}$$
$$=A\cos{kx}+B\sin{kx}$$
where ##c_1, c_2, A,## and ##B## are complex constants.
We find ##A## and ##B## by applying boundary conditions.
Our boundary conditions arise from assuming that ##\psi(x)## is continuous at ##x=0## and ##x=l##.
To be continuous at ##x=0## it must be that the limits of the wavefunction approaching ##0## from left and right are the same.
$$\lim\limits_{x\to 0}\psi_{I}=\lim\limits_{x\to 0}\psi_{II}$$
$$0=\lim\limits_{x\to 0}\left [A\cos{\left ( \frac{\sqrt{2mE}x}{\hbar} \right )}+B\sin{\left ( \frac{\sqrt{2mE}x}{\hbar} \right )}\right ]$$
$$=A$$
Thus
$$\psi_{II}=B\sin{\left ( \frac{\sqrt{2mE}x}{\hbar} \right )}\tag{2}$$
My question is about the calculations involving the second boundary condition, namely that the wavefunction is continuous at ##x=l##.
$$\lim\limits_{x\to l}\psi_{III}=\lim\limits_{x\to l}\psi_{II}$$
$$0=\lim\limits_{x\to l}\left [B\sin{\left ( \frac{\sqrt{2mE}x}{\hbar} \right )}\right ]$$
$$B\sin{\left ( kl \right )}=0$$
$$\implies kl = n\pi, n\in \mathbb{N}\tag{3}$$
Okay, notice the ##n\in\mathbb{N}## in (3). Why do we not say ##n\in\mathbb{Z}##?
Well, if ##n<0## then we get essentially the same solutions (and I think the only difference is the sign on the energies that we get).
My question is, however, about the case of ##n=0##.
If ##n=0## then at first glance it would seem that ##k=0## and from (2) we would have ##\psi_{II}=0##.
Can we just conclude this directly like this?
Another way to reach this conclusion is to look at the original differential equation. If ##n=0## then
$$\frac{\partial^2\psi_{II}(x)}{\partial x^2}+k^2\psi_{II}(x)=0$$
$$\frac{\partial^2\psi_{II}(x)}{\partial x^2}=0$$
$$\implies \psi(x)=ax+b$$
which is a solution that only obeys both boundary conditions if ##a=b=0##.
Thus, it must be that ##\psi(x)=0##.
Is it ok to simply insert ##n=0## into (2) and from there conclude that ##\psi(x)=0##?
For context about where in the calculations I am, we assume that a particle is subjected to a potential energy function that is infinite everywhere along the ##x##-axis except for a line segment of length ##l## where the potential energy is zero.
Thus, the ##x##-axis is divided into three regions where we will solve Schrödinger's equation
It can be shown that in regions I and III, the solution to (1) is just ##\psi(x)=0##.
Consider region II.
$$-\frac{\hbar^2}{2m}\frac{\partial^2\psi_{II}(x)}{\partial x^2}=\psi_{II}(x)E$$
$$\frac{\partial^2\psi_{II}(x)}{\partial x^2}=-\frac{2m}{\hbar^2}E\psi_{II}(x)$$
$$\frac{\partial^2\psi_{II}(x)}{\partial x^2}+\frac{2mE}{\hbar^2}\psi_{II}(x)=0$$
$$\frac{\partial^2\psi_{II}(x)}{\partial x^2}+k^2\psi_{II}(x)=0$$
This is a 2nd-order, linear, homogeneous ODE with
$$k^2=\frac{2mE}{\hbar^2}$$
A general (complex) solution is
$$\psi_{II}=c_1e^{x\frac{i\sqrt{2mE}}{\hbar}}+c_2e^{\frac{-ix\sqrt{2mE}}{\hbar}}$$
$$=c_1e^{ikx}+c_2e^{-ikx}$$
$$=c_1(\cos(kx)+i\sin{kx})+c_2(\cos{kx}-i\sin{kx})$$
$$=(c_1+c_2)\cos{kx}+i(c_1-c_2)\sin{kx}$$
$$=A\cos{kx}+B\sin{kx}$$
where ##c_1, c_2, A,## and ##B## are complex constants.
We find ##A## and ##B## by applying boundary conditions.
Our boundary conditions arise from assuming that ##\psi(x)## is continuous at ##x=0## and ##x=l##.
To be continuous at ##x=0## it must be that the limits of the wavefunction approaching ##0## from left and right are the same.
$$\lim\limits_{x\to 0}\psi_{I}=\lim\limits_{x\to 0}\psi_{II}$$
$$0=\lim\limits_{x\to 0}\left [A\cos{\left ( \frac{\sqrt{2mE}x}{\hbar} \right )}+B\sin{\left ( \frac{\sqrt{2mE}x}{\hbar} \right )}\right ]$$
$$=A$$
Thus
$$\psi_{II}=B\sin{\left ( \frac{\sqrt{2mE}x}{\hbar} \right )}\tag{2}$$
My question is about the calculations involving the second boundary condition, namely that the wavefunction is continuous at ##x=l##.
$$\lim\limits_{x\to l}\psi_{III}=\lim\limits_{x\to l}\psi_{II}$$
$$0=\lim\limits_{x\to l}\left [B\sin{\left ( \frac{\sqrt{2mE}x}{\hbar} \right )}\right ]$$
$$B\sin{\left ( kl \right )}=0$$
$$\implies kl = n\pi, n\in \mathbb{N}\tag{3}$$
Okay, notice the ##n\in\mathbb{N}## in (3). Why do we not say ##n\in\mathbb{Z}##?
Well, if ##n<0## then we get essentially the same solutions (and I think the only difference is the sign on the energies that we get).
My question is, however, about the case of ##n=0##.
If ##n=0## then at first glance it would seem that ##k=0## and from (2) we would have ##\psi_{II}=0##.
Can we just conclude this directly like this?
Another way to reach this conclusion is to look at the original differential equation. If ##n=0## then
$$\frac{\partial^2\psi_{II}(x)}{\partial x^2}+k^2\psi_{II}(x)=0$$
$$\frac{\partial^2\psi_{II}(x)}{\partial x^2}=0$$
$$\implies \psi(x)=ax+b$$
which is a solution that only obeys both boundary conditions if ##a=b=0##.
Thus, it must be that ##\psi(x)=0##.
Is it ok to simply insert ##n=0## into (2) and from there conclude that ##\psi(x)=0##?