rlm@0: #+TITLE: Bugs in Quantum Mechanics rlm@0: #+AUTHOR: Dylan Holmes rlm@0: #+SETUPFILE: ../../aurellem/org/setup.org rlm@0: #+INCLUDE: ../../aurellem/org/level-0.org rlm@0: rlm@0: rlm@0: #Bugs in the Quantum-Mechanical Momentum Operator rlm@0: rlm@0: rlm@0: I studied quantum mechanics the same way I study most subjects\mdash{} rlm@0: by collecting (and squashing) bugs in my understanding. One of these rlm@0: bugs persisted throughout two semesters of rlm@0: quantum mechanics coursework until I finally found rlm@0: the paper rlm@0: [[http://arxiv.org/abs/quant-ph/0103153][/Self-adjoint extensions of operators and the teaching of quantum rlm@0: mechanics/]], which helped me stamp out the bug entirely. I decided to rlm@0: write an article about the problem and its solution for a number of reasons: rlm@0: rlm@0: - Although the paper was not unreasonably dense, it was written for rlm@0: teachers. I wanted to write an article for students. rlm@0: - I wanted to popularize the problem and its solution because other rlm@0: explanations are currently too hard to find. (Even Shankar's rlm@0: excellent [[http://books.google.com/books/about/Principles_of_quantum_mechanics.html?id=2zypV5EbKuIC][textbook]] doesn't mention it.) rlm@0: - I wanted to check that the bug was indeed entirely rlm@0: eradicated. Attempting an explanation is my way of making rlm@0: sure. rlm@0: rlm@0: * COMMENT rlm@0: I recommend the rlm@0: paper not only for students who are learning rlm@0: quantum mechanics, but especially for teachers interested in debugging rlm@0: them. rlm@0: rlm@0: * COMMENT rlm@0: On my first exam in quantum mechanics, my professor asked us to rlm@0: describe how certain measurements would affect a particle in a rlm@0: box. Many of these measurement questions required routine application rlm@0: of skills we had recently learned\mdash{}first, you recall (or rlm@0: calculate) the eigenstates of the quantity rlm@0: to be measured; second, you write the given state as a linear rlm@0: sum of these eigenstates\mdash{} the coefficients on each term give rlm@0: the probability amplitude. rlm@0: rlm@0: * The infinite square well potential rlm@0: rlm@0: There is a particle in a one-dimensional potential well that is rlm@0: infinite everywhere except for a well of length \(a\). This means that the rlm@0: particle exists in a potential[fn:coords][fn:infinity] rlm@0: rlm@0: rlm@0: \(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for rlm@0: }\;x<0\text{ or }x>a.\end{cases}\) rlm@0: rlm@0: The Schr\ouml{}dinger equation describes how the particle's state rlm@0: \(|\psi\rangle\) will change over time in this system. rlm@0: rlm@0: \(\begin{eqnarray} rlm@0: i\hbar \frac{\partial}{\partial t}|\psi\rangle &=& rlm@0: H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\) rlm@0: rlm@0: This is a differential equation whose solutions are the physically rlm@0: allowed states for the particle in this system. Physically allowed rlm@0: states are those that change in physically allowed ways. Like any rlm@0: differential equation, the Schr\ouml{}dinger equation can be rlm@0: accompanied by /boundary conditions/\mdash{}conditions that rlm@0: further restrict which states qualify as physically allowed. rlm@0: rlm@0: Whenever possible, physicists impose these boundary conditions: rlm@0: - The state should be a /continuous function of/ \(x\). This means rlm@0: that if a particle is very likely to be /at/ a particular location, rlm@0: it is also very likely to be /near/ that location. rlm@0: - rlm@0: rlm@0: #; physically allowed states are those that change in physically rlm@0: #allowed ways. rlm@0: rlm@0: rlm@0: ** Boundary conditions rlm@0: Because the potential is infinite everywhere except within the well, rlm@0: a realistic particle must be confined to exist only within the rlm@0: well\mdash{}its wavefunction must be zero everywhere beyond the walls rlm@0: of the well. rlm@0: rlm@0: rlm@0: [fn:coords] I chose my coordinate system so that the well extends from rlm@0: \(0