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: #Bugs in Quantum Mechanics 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: rlm@0: * What I thought I knew rlm@0: rlm@0: The following is a list of things I thought were true of quantum rlm@0: mechanics; the catch is that the list contradicts itself. rlm@0: rlm@0: - For any hermitian operator: Eigenstates with different eigenvalues are orthogonal. rlm@0: - For any hermitian operator: Any physically allowed state can be rlm@0: written as a linear sum of eigenstates of the operator. rlm@0: - The momentum operator and energy operator are hermitian, because rlm@0: momentum and energy are measureable quantities. rlm@0: - In vacuum, rlm@0: - the momentum operator has an eigenstate rlm@0: \(p(x)=\exp{(ipx/\hbar)}\) for each value of $p$. rlm@0: - the energy operator has an eigenstate \(|E\rangle = rlm@0: \alpha|p\rangle + \beta|-p\rangle\) for any \(\alpha,\beta\) and rlm@0: the particular choice of momentum $p=\sqrt{2mE}$. rlm@0: - In the infinitely deep potential well, rlm@0: - the momentum operator has eigenstates with the same form $p(x) = rlm@0: \exp{(ipx/\hbar)}$, but because of the boundary conditions on the rlm@0: well, the following modifications are required. rlm@0: - The wavefunction must be zero everywhere outside the well. That rlm@0: is, \(p(x) = \begin{cases}\exp{(ipx/\hbar)},& 0\lt{}x\lt{}a; rlm@0: \\0, & \text{for }x<0\text{ or }x>a \\ \end{cases}\) rlm@0: #0,&\text{for }x\lt{}0\text{ or }x\gt{}a\end{cases}\) rlm@0: - no longer has an eigenstate for each value rlm@0: of $p$. Instead, only values of $p$ that are integer multiples of rlm@0: $\pi a/\hbar$ are physically realistic. rlm@0: rlm@0: rlm@0: rlm@0: * COMMENT: rlm@0: rlm@0: ** Eigenstates with different eigenvalues are orthogonal rlm@0: rlm@0: #+begin_quote rlm@0: *Theorem:* Eigenstates with different eigenvalues are orthogonal. rlm@0: #+end_quote rlm@0: rlm@0: ** COMMENT : rlm@0: I can prove this: if $\Lambda$ is any linear operator, suppose $|a\rangle$ rlm@0: and $|b\rangle$ are eigenstates of $\Lambda$. This means that rlm@0: rlm@0: rlm@0: \( rlm@0: \begin{eqnarray} rlm@0: \Lambda |a\rangle&=& a|a\rangle,\\ rlm@0: \Lambda|b\rangle&=& b|b\rangle.\\ rlm@0: \end{eqnarray} rlm@0: \) rlm@0: rlm@0: If we take the difference of these eigenstates, we find that rlm@0: rlm@0: \( rlm@0: \begin{eqnarray} rlm@0: \Lambda\;\left(|a\rangle-|b\rangle\right) &=& \Lambda |a\rangle - \Lambda |b\rangle rlm@0: \qquad \text{(because $\Lambda$ is linear.)}\\ rlm@0: &=& a|a\rangle - b|b\rangle\qquad\text{(because $|a\rangle$ and rlm@0: $|b\rangle$ are eigenstates of $\Lambda$)} rlm@0: \end{eqnarray}\) rlm@0: rlm@0: rlm@0: which means that $a\neq b$. rlm@0: rlm@0: ** Eigenvectors of hermitian operators span the space of solutions rlm@0: rlm@0: #+begin_quote rlm@0: *Theorem:* If $\Omega$ is a hermitian operator, then every physically rlm@0: allowed state can be written as a linear sum of eigenstates of rlm@0: $\Omega$. rlm@0: #+end_quote rlm@0: rlm@0: rlm@0: rlm@0: ** Momentum and energy are hermitian operators rlm@0: This ought to be true because hermitian operators correspond to rlm@0: observable quantities. Since we expect momentum and energy to be rlm@0: measureable quantities, we expect that there are hermitian operators rlm@0: to represent them. rlm@0: rlm@0: rlm@0: ** Momentum and energy eigenstates in vacuum rlm@0: An eigenstate of the momentum operator $P$ would be a state rlm@0: \(|p\rangle\) such that \(P|p\rangle=p|p\rangle\). rlm@0: rlm@0: ** Momentum and energy eigenstates in the infinitely deep well rlm@0: rlm@0: rlm@0: rlm@0: * Can you measure momentum in the infinite square well? rlm@0: rlm@0: rlm@0: rlm@0: ** COMMENT Momentum eigenstates rlm@0: rlm@0: In free space, the Hamiltonian is \(H=\frac{1}{2m}P^2\) and the rlm@0: momentum operator $P$ has eigenstates \(p(x) = \exp{(-ipx/\hbar)}\). rlm@0: rlm@0: In the infinitely deep potential well, the Hamiltonian is the same but rlm@0: there is a new condition in order for states to qualify as physically rlm@0: allowed: the states must not exist anywhere outside of well, as it rlm@0: takes an infinite amount of energy to do so. rlm@0: rlm@0: Notice that the momentum eigenstates defined above do /not/ satisfy rlm@0: this condition. rlm@0: rlm@0: rlm@0: rlm@0: * COMMENT rlm@0: For each physical system, there is a Schr\ouml{}dinger equation that rlm@0: describes how a particle's state $|\psi\rangle$ will change over rlm@0: time. 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; each solution to the rlm@0: Schr\ouml{}dinger equation is a state that is physically allowed for rlm@0: our particle. Here, physically allowed states are rlm@0: those that change in physically allowed ways. However, like any differential rlm@0: equation, the Schr\ouml{}dinger equation can be accompanied by rlm@0: /boundary conditions/\mdash{}conditions that further restrict which rlm@0: states qualify as physically allowed. rlm@0: rlm@0: rlm@0: rlm@0: rlm@0: ** Eigenstates of momentum rlm@0: rlm@0: rlm@0: rlm@0: rlm@0: #In the infinitely deep well potential $V(x)=0$, the Schr\ouml{}dinger rlm@0: rlm@0: #\(i\hbar\frac{\partial}{\partial t}|\psi\rangle = H|\psi\rangle\) rlm@0: rlm@0: rlm@0: rlm@0: rlm@0: rlm@0: rlm@0: rlm@0: * COMMENT rlm@0: rlm@0: #* The infinite square well potential rlm@0: rlm@0: A particle exists in a potential that is rlm@0: infinite everywhere except for a region of length \(a\), where the potential is zero. 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; each solution to the rlm@0: Schr\ouml{}dinger equation is a state that is physically allowed for rlm@0: our particle. Here, physically allowed states are rlm@0: those that change in physically allowed ways. However, like any differential rlm@0: equation, the Schr\ouml{}dinger equation can be accompanied by rlm@0: /boundary conditions/\mdash{}conditions that further restrict which rlm@0: states qualify as physically allowed. rlm@0: rlm@0: rlm@0: Whenever possible, physicists impose these boundary conditions: rlm@0: - A physically allowed state ought to be a /smoothly-varying function of position./ This means rlm@0: that if a particle in the state is likely to be /at/ a particular location, rlm@0: it is also likely to be /near/ that location. rlm@0: rlm@0: These boundary conditions imply that for the square well potential in rlm@0: this problem, rlm@0: rlm@0: - Physically allowed states must be totally confined to the well, rlm@0: because it takes an infinite amount of energy to exist anywhere rlm@0: outside of the well (and physically allowed states ought to have rlm@0: only finite energy). rlm@0: - Physically allowed states must be increasingly unlikely to find very rlm@0: close to the walls of the well. This is because of two conditions: the above rlm@0: condition says that the particle is /impossible/ to find rlm@0: outside of the well, and the smoothly-varying condition says rlm@0: that if a particle is impossible to find at a particular location, rlm@0: it must be unlikely to be found nearby that location. 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