#+TITLE: Bugs in Quantum Mechanics

 #+AUTHOR: Dylan Holmes

 #+SETUPFILE: ../../aurellem/org/setup.org

 #+INCLUDE:   ../../aurellem/org/level-0.org

 

 

 #Bugs in the Quantum-Mechanical Momentum Operator

 

 

 I studied quantum mechanics the same way I study most subjects\mdash{}

 by collecting (and squashing) bugs in my understanding. One of these

 bugs persisted throughout two semesters of

 quantum mechanics coursework until I finally found

 the paper 

 [[http://arxiv.org/abs/quant-ph/0103153][/Self-adjoint extensions of operators and the teaching of quantum

 mechanics/]], which helped me stamp out the bug entirely. I decided to

 write an article about the problem and its solution for a number of reasons:

 

 - Although the paper was not unreasonably dense, it was written for

   teachers. I wanted to write an article for students.

 - I wanted to popularize the problem and its solution because other

   explanations are currently too hard to find. (Even Shankar's

   excellent [[http://books.google.com/books/about/Principles_of_quantum_mechanics.html?id=2zypV5EbKuIC][textbook]] doesn't mention it.)

 - I wanted to check that the bug was indeed entirely

   eradicated. Attempting an explanation is my way of making

   sure.

 

 * COMMENT

  I recommend the

 paper not only for students who are learning

 quantum mechanics, but especially for teachers interested in debugging

 them. 

 

 * COMMENT

 On my first exam in quantum mechanics, my professor asked us to

 describe how certain measurements would affect a particle in a

 box. Many of these measurement questions required routine application

 of skills we had recently learned\mdash{}first, you recall (or

 calculate) the eigenstates of the quantity

 to be measured; second, you write the given state as a linear

 sum of these eigenstates\mdash{} the coefficients on each term give

 the probability amplitude.

 

 * The infinite square well potential

 

 There is a particle in a one-dimensional potential well that is

 infinite everywhere except for a well of length \(a\). This means that the

 particle exists in a potential[fn:coords][fn:infinity]

 

 

 \(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for

 }\;x<0\text{ or }x>a.\end{cases}\)

 

 The Schr\ouml{}dinger equation describes how the particle's state 

 \(|\psi\rangle\) will change over time in this system.

 

 \(\begin{eqnarray}

 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&

 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)

 

 This is a differential equation whose solutions are the physically

 allowed states for the particle in this system. Physically allowed

 states are those that change in physically allowed ways. Like any

 differential equation, the Schr\ouml{}dinger equation can be

 accompanied by /boundary conditions/\mdash{}conditions that

 further restrict which states qualify as physically allowed.

 

 Whenever possible, physicists impose these boundary conditions:

 - The state should be a /continuous function of/ \(x\). This means

   that if a particle is very likely to be /at/ a particular location,

   it is also very likely to be /near/ that location.

 - 

 

 #; physically allowed states are those that change in physically

 #allowed ways.

 

 

 ** Boundary conditions

 Because the potential is infinite everywhere except within the well,

 a realistic particle must be confined to exist only within the

 well\mdash{}its wavefunction must be zero everywhere beyond the walls

 of the well.

 

 

 [fn:coords] I chose my coordinate system so that the well extends from

 \(0<x<a\). Others choose a coordinate system so that the well extends from

 \(-\frac{a}{2}<x<\frac{a}{2}\). Although both coordinate systems describe the same physical

 situation, they give different-looking answers.

 

 [fn:infinity] Of course, infinite potentials are not

 realistic. Instead, they are useful approximations to finite

 potentials when \ldquo{}infinity\rdquo{} and \ldquo{}the actual height

 of the well\rdquo{} are close enough for your own practical

 purposes. Having introduced a physical impossibility into the problem

 already, we don't expect to get physically realistic solutions; we

 just expect to get mathematically consistent ones. The forthcoming

 trouble is that we don't.