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PHY356F: Quantum Mechanics I. Lecture 9 — Bound states.

Posted by peeterjoot on November 16, 2010

My notes from Lecture 9, November 16, 2010. Taught by Prof. Vatche Deyirmenjian.

[Click here for a PDF of this post with nicer formatting]

Motivation. Motivation for today’s physics is Solar Cell technology and quantum dots.

Problem:

What ware the eigenvalues and eigenvectors for an electron trapped in a 1D potential well?

MODEL.

Quantum state \lvert {\Psi} \rangle describes the particle. What V(X) should we choose? Try a quantum well with infinite barriers first.

These spherical quantum dots are like quantum wells. When you trap electrons in this scale you’ll get energy quantization.

VISUALIZE.

Draw a picture for V(X) with infinite spikes at \pm a. (ie: figure 8.1 in the text).

SOLVE.

First task is to solve the time independent Schr\”{o}dinger equation.

\begin{aligned}H {\lvert {\Psi} \rangle} = E {\lvert {\Psi} \rangle}\end{aligned} \hspace{\stretch{1}}(5.98)

derivable from

\begin{aligned}H {\lvert {\Psi} \rangle} = i \hbar \frac{\partial {}}{\partial {t}} {\lvert {\Psi} \rangle}\end{aligned} \hspace{\stretch{1}}(5.99)

In the position representation, we project {\langle {x} \rvert} onto H {\lvert {\Psi} \rangle} and solve for \left\langle{{x}} \vert {{\Psi}}\right\rangle = \Psi(x). For the problems in Chapter 8,

\begin{aligned}H = \frac{\mathbf{P}^2}{2m} + V(X,Y,Z),\end{aligned} \hspace{\stretch{1}}(5.100)

where

\begin{aligned}P &= \text{momentum operator} \\ X &= \text{position operator} \\ m &= \text{electron mass}\end{aligned}

We should be careful to be strict about the notation, and not interchange the operators and their specific representations (ie: not interchanging “little-x” and “big-x”) as we see in the text in this chapter.

Here the potential energy operator V(X,Y,Z) is time independent.

If i \hbar \frac{d{\lvert {\Psi} \rangle}}{dt} = H {\lvert {\Psi} \rangle} and H is time independent then {\lvert {\Psi} \rangle} = {\lvert {u} \rangle} e^{-i E t/\hbar} implies

\begin{aligned}i \hbar \frac{ -i E }{\hbar} {\lvert {u} \rangle} e^{-i E t/\hbar} = H {\lvert {u} \rangle} e^{-i E t/\hbar}\end{aligned}

or

\begin{aligned}E {\lvert {u} \rangle} = H {\lvert {u} \rangle}\end{aligned} \hspace{\stretch{1}}(5.101)

Here E is the energy eigenvalue, and {\lvert {u} \rangle} is the energy eigenstate. Our differential equation now becomes

\begin{aligned}-\frac{\hbar^2 }{2m} \frac{d^2 u(x)}{dx^2} + V(x) u(x) = E u(x)\end{aligned} \hspace{\stretch{1}}(5.102)

where V(x) = 0 for {\left\lvert{x}\right\rvert} < a. We won't find anything like this for real, but this is our first approximation to the quantum dot.

Our differential equation in the well is now

\begin{aligned}-\frac{\hbar^2 }{2m} \frac{d^2 u(x)}{dx^2} = E u(x)\end{aligned} \hspace{\stretch{1}}(5.103)

or with \alpha = \sqrt{2m E/\hbar^2}

\begin{aligned}\frac{d^2 u(x)}{dx^2} u(x) = -\frac{2 m E}{\hbar^2} u(x) = - \alpha^2 u(x)\end{aligned} \hspace{\stretch{1}}(5.104)

Our solution for {\left\lvert{x}\right\rvert} < a is then

\begin{aligned}u(x) = A \cos \alpha x + B \sin\alpha x\end{aligned} \hspace{\stretch{1}}(5.105)

and for {\left\lvert{x}\right\rvert} > a we have u(x) = 0 since V(x) = \infty.

Setting u(a) = u(-a) = 0 we have

\begin{aligned}A \cos \alpha a + B \sin\alpha a &= 0 \\ A \cos \alpha a - B \sin\alpha a &= 0\end{aligned}

Type I.

B=0, A \cos\alpha a = 0. For A \ne 0 we must have

\begin{aligned}\cos \alpha a = 0\end{aligned}

or \alpha a = n \frac{\pi}{2}, where n = 1, 3, 5, ..., so our solution is

\begin{aligned}u(x) = A \cos \left( \frac{n \pi}{2 a} x \right) \end{aligned} \hspace{\stretch{1}}(5.106)

Type II.

A=0, B \sin\alpha a = 0. For B \ne 0 we must have

\begin{aligned}\sin \alpha a = 0\end{aligned}

or \alpha a = n \frac{\pi}{2}, where n = 1, 2, 4, ..., so our solution is

\begin{aligned}u(x) = B \sin \left( \frac{n \pi}{2 a} x \right) \end{aligned} \hspace{\stretch{1}}(5.107)

Via determinant

We could also write

\begin{aligned}\begin{bmatrix}\cos \alpha a & \sin\alpha a \\ \cos \alpha a & - \sin\alpha a \end{bmatrix}\begin{bmatrix}A \\ B\end{bmatrix}= 0\end{aligned}

and then must have zero determinant, or

\begin{aligned}-2 \sin\alpha a \cos\alpha a = -\sin 2 \alpha a\end{aligned} \hspace{\stretch{1}}(5.108)

so we must have

\begin{aligned}2 \alpha a = n \pi\end{aligned}

or

\begin{aligned}\alpha = \frac{n \pi}{2a}\end{aligned}

regardless of A and B. We can then determine the solutions 5.106, and 5.107 simply by noting that this value for \alpha kills off either the sine or cosine terms of 5.105 depending on whether n is even or odd.

CHECK.

\begin{aligned}u_n(x) &= A \cos \left( \frac{n \pi}{2 a} x \right) \\ u_n(x) &= B \sin \left( \frac{n \pi}{2 a} x \right) \end{aligned}

satisfy the time independent Schr\”{o}dinger equation, and the corresponding eigenvalues from from

\begin{aligned}\alpha = \sqrt{\frac{2 m E}{\hbar^2}},\end{aligned}

or

\begin{aligned}E = \frac{\hbar^2 \alpha^2}{2m} = \frac{\hbar^2}{2m} \left( \frac{n \pi}{2a} \right)^2 \end{aligned}

for n = 1, 2, 3, \cdots.

On the derivative of u at the boundaries

Integrating

\begin{aligned}-\frac{\hbar^2 }{2m} \frac{d^2 u(x)}{dx^2} u(x) + V(x) u(x) = E u(x),\end{aligned} \hspace{\stretch{1}}(5.109)

over [a-\epsilon,a+\epsilon] we have

\begin{aligned}-\frac{\hbar^2 }{2m} &\int_{a-\epsilon}^{a-\epsilon}\frac{d^2 u(x)}{dx^2} dx+ \int_{a-\epsilon}^{a-\epsilon}V(x) u(x) dx = \int_{a-\epsilon}^{a-\epsilon}E u(x) dx \\ -\frac{\hbar^2 }{2m} &\left( \left.\frac{du}{dx}\right\vert_{a-\epsilon}^{a+\epsilon} + 0 = 0\right)\end{aligned} \hspace{\stretch{1}}(5.110)

which gives us

\begin{aligned}\left.\frac{du}{dx}\right\vert_{a + \epsilon}-\left.\frac{du}{dx}\right\vert_{a - \epsilon} = 0\end{aligned} \hspace{\stretch{1}}(5.112)

or

\begin{aligned}\left.\frac{du}{dx}\right\vert_{a + \epsilon}&=\left.\frac{du}{dx}\right\vert_{a - \epsilon} \end{aligned} \hspace{\stretch{1}}(5.113)

We can infer how the derivative behaves over the potential discontinuity, so in the limit where \epsilon \rightarrow 0 we must have wave function continuity at despite the potential discontinuity.

This sort of analysis, which is potential dependent, we see that for this infinite well potential, our derivative must be continuous at the boundary.

Problem:

non-infinite step well potential.

Given a zero potential in the well {\left\lvert{x}\right\rvert} < a

\begin{aligned}-\frac{\hbar^2 }{2m} \frac{d^2 u(x)}{dx^2} u(x) + 0 = E u(x),\end{aligned} \hspace{\stretch{1}}(5.114)

and outside of the well {\left\lvert{x}\right\rvert} > a

\begin{aligned}-\frac{\hbar^2 }{2m} \frac{d^2 u(x)}{dx^2} u(x) + V_0 u(x) = E u(x)\end{aligned} \hspace{\stretch{1}}(5.115)

Inside of the well, we have the solution worked previously, with \alpha = \sqrt{2m E/\hbar^2}

\begin{aligned}u(x) &= A \cos\alpha x + B \sin\alpha x \end{aligned} \hspace{\stretch{1}}(5.116)

Then we have outside of the well the same form

\begin{aligned}-\frac{\hbar^2 }{2m} \frac{d^2 u(x)}{dx^2} u(x) = (E - V_0 )u(x) \end{aligned} \hspace{\stretch{1}}(5.117)

With \beta = \sqrt{ 2m (V_0 - E)/\hbar^2}, this is

\begin{aligned}\frac{d^2 u(x)}{dx^2} u(x) = \beta^2 u(x) \end{aligned} \hspace{\stretch{1}}(5.118)

If V_0 - E > 0, we have V_0 > E, and the states are “bound” or “localized” in the well.

Our solutions for this V_0 > E case are then

\begin{aligned}u(x) &= D e^{\beta x} \\ u(x) &= C e^{-\beta x}\end{aligned} \hspace{\stretch{1}}(5.119)

for x \le a, and x \ge a respectively.

Question: Why can we not have

\begin{aligned}u(x) = D e^{\beta x} + C e^{-\beta x}\end{aligned} \hspace{\stretch{1}}(5.121)

for x \le -a?

Answer: As x \rightarrow -\infty we would then have

\begin{aligned}u(x) \rightarrow C e^{\beta \infty} \rightarrow \infty\end{aligned}

This is a non-physical solution, and we discard it based on our normalization requirement.

Our total solution, in regions x  a respectively

\begin{aligned}u_1(x) &= D e^{\beta x} \\ u_2(x) &= A \cos\alpha x + B \sin\alpha x \\ u_3(x) &= C e^{-\beta x}\end{aligned}

To find the coefficients, set u_1(-a) = u_2(-a), u_2(a) = u_3(a) u_1'(-a) = u_2'(-a), u_2'(a) = u_3'(a), and NORMALIZE u(x).

Now, how about in region 2 (x < -a), V_0 < E implies that our equation is

\begin{aligned}\frac{d^2 u(x)}{dx^2} u(x) = - \frac{2m}{\hbar^2} (E - V_0) u(x) = - k^2 u(x)\end{aligned} \hspace{\stretch{1}}(5.122)

We no longer have quantized energy for such a solution. These correspond to the “unbound” or “continuum” states. Even though we do not have quantized energy we still have quantum effects. Our solution becomes

\begin{aligned}u_1(x) &= C_2 e^{i k x} +D_2 e^{-i k x}  \\ u_2(x) &= A e^{i \alpha x} +B e^{-i \alpha x}  \\ u_3(x) &= C_3 e^{i k x} \end{aligned}

Question. Why no D_2 e^{-i k x}, in the u_3(x) term?

Answer. We can, but this is not physically relevant. Why is because we associate e^{ikx} with an incoming wave, with reflection in the x < -a interval, and both e^{\pm i \alpha x} in the $latex {\left\lvert{x}\right\rvert} a$ region.

FIXME: scan picture: 9.1 in my notebook.

Observe that this is not normalizable as is. We require “delta-function” normalization. What we can do is ask about current densities. How much passes through the barrier, and so forth.

Note to self. We probably really we want to consider a wave packet of states, something like:

\begin{aligned}\Psi_1(x) &= \int dk f_1(k) e^{i k x} \\ \Psi_2(x) &= \int d\alpha f_2(\alpha) e^{i \alpha x} \\ \Psi_3(x) &= \int dk f_3(k) e^{i k x}\end{aligned}

Then we’d have something that we can normalize. Play with this later.

Setup for next week’s hydrogen atom lecture.

We’ll want to solve this using the formalism we’ve discussed. The general problem is a proton, positively charged, with a nearby negative charge (the electron).

Our equation to solve is

\begin{aligned}\left(-\frac{\hbar^2}{2 m_1} \boldsymbol{\nabla}_1^2-\frac{\hbar^2}{2 m_2} \boldsymbol{\nabla}_2^2\right)u(\mathbf{r}_1, \mathbf{r}_2) + V(\mathbf{r}_1, \mathbf{r}_2)u(\mathbf{r}_1, \mathbf{r}_2)=E u(\mathbf{r}_1, \mathbf{r}_2).\end{aligned} \hspace{\stretch{1}}(6.123)

Here \left( -\frac{\hbar^2}{2 m_1} \boldsymbol{\nabla}_1^2 -\frac{\hbar^2}{2 m_2} \boldsymbol{\nabla}_2^2 \right) is the total kinetic energy term. For hydrogen we can consider the potential to be the Coulomb potential energy function that depends only on \mathbf{r}_1 - \mathbf{r}_2. We can transform this using a center of mass transformation. Introduce the centre of mass coordinate and relative coordinate vectors

\begin{aligned}\mathbf{R} &= \frac{m_1 \mathbf{r}_1 + m_2 \mathbf{r}_2}{ m_1 + m_2 } \\ \mathbf{r} &= \mathbf{r}_1 - \mathbf{r}_2\end{aligned} \hspace{\stretch{1}}(6.124)

With this transformation we can reduce the problem to a single coordinate PDE.

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