#+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 has

 infinitely high walls and finite width \(a\). 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. Like any differential

 equation, 

 

 

 Like any differential equation, the Schr\ouml{}dinger equation 

 #; 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.