Responsible party: Zeno, Blackbox

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• Special Functions

• Raising/Lowering Operators

• Structuring DEs

• Expectation of Momentum / Momentum Squared

• Orthogonality of Hermitian Operators

• Introduction to the Analytical Method (next lecture)

One of the first points made today was that special functions such as the Bessel Functions, Legendre Polynomials and Hermite Polynomials keep emerging as solutions to various classes of differential equations. The important aspect of this fact is that when one function of a class is a solution, usually other functions in that class and sometimes linear combinations of the functions are solutions as well. One of the most familiar examples is the types of differential equations we commonly see as physics students: <math> y + k^2 y = 0</math>, <math> y +b y' +k^2 y = 0</math>, etc., which have exponential solutions. Similar DEs have other exponential solutions, and using Euler's Formulae we can express these as trigonometric functions; all being a class of exponential functions that form solutions to these types of differential equations. This concept pertains to Quantum Mechanics with the exponential and trig solutions we've used thus far, and with the Hermite Polynomials in the Analytical solution for the wave functions of the Harmonic Oscillator (see pg 56), which we'll cover Wednesday.

As derived and discussed in the last lecture as the mechanism for the Algebraic Method of solution to the Harmonic Oscillator, the ladder operators are:

<math> a_\pm = \frac{1}{\sqrt{2 \hbar m \omega}}(\mp i p + m \omega x)</math> for the raising and lowering operator, respectively. Notice that the term outside the parentheses makes the ladder operators dimensionless quantities: <math>\frac{1}{\sqrt{(kg*m^2/s)(kg)(1/s)}}(kg*m/s)=dimensionless</math>

• It is often helpful and preferable to structure factors in DEs and operations such as the ladder operators as dimensionless quantities.

A few more examples are: <math>\xi=x/x_0</math> is natural length scale and

<math> E/E_0</math> is a natural energy scale for <math> E_0 = \hbar \omega </math> for the Harmonic Oscillator.

Recall that we can write momentum as <math> p=-i\hbar\partial_x </math>

⇒ then the Hamiltonian is <math> H=\hbar\omega(\partial_\xi^2+ \xi^2) </math>

and the ladder operators become <math> a_\pm= \mp \partial_\xi+\xi </math>

Rewritten with the basic ladder operators, The Schrodinger equation is: <math> \hbar\omega(a_+a_- + 1/2)\psi_n=E_n \psi_n = \hbar \omega(n+1/2) \psi_n</math>

and canceling, <math>a_+a_-\psi_n=n\psi_n</math>

• Even though the ladder operators seem very complicated, using characteristics like their relation to the Hamiltonian and commutation, they prove to be very useful and probably easier and simpler than the Analytical approach to the Harmonic Oscillator.

The Ladder Operators allow us to write the Hamiltonian as <math> H=\hbar\omega(a_+a_- + 1/2) </math>.

and the Schrodinger equation as <math> \hbar\omega(a_\mp a_\pm \pm (1/2))\psi=E\psi </math>

It was asked in lecture today how to compute the expectation value of the momentum and momentum^2. Yuichi showed us that <math> <p>_\Psi_n = \int_{-\infty}^{\infty}\Psi^*(x,t)(-i \hbar \partial_x)\Psi(x,t)dx</math>

which yields <math>\Psi_0=(\frac{m\omega}{\pi\hbar})^{1/4}e^{-\xi^2/2}</math> for n = 0

and <math>\Psi_n=sqrt{2/L}sin(n \pi x / L)</math> for <math>n=0=\Psi_0(x)==0</math>

⇒ <math> \Psi_n(x) = sqrt{2/L}sin(\frac{n+1}{L} \pi x) </math> for <math> n >= 0 </math>.

*Starting at n=0 or n=1 is purely a matter of convention. Equations like

<math>E_n=\hbar\omega(n+1/2)</math> for <math>n>=0</math>

and <math>E_n=\hbar\omega(n-1/2)</math> for <math>n>=1</math>

are purely based on notational preferences and chosen conventions, and are identical physically.

Now the expectation value of momentum is given by

<math> <p>_\Psi_n = \int_{-\infty}^{\infty}\Psi_n^*(-i sqrt{\frac{m \hbar \omega}{2}} (a_- - a_+))\Psi_n dx</math>

<math> = (-i sqrt{\frac{m \hbar \omega}{2}}) \int_{-\infty}^{\infty}(\Psi_n^*(a_-)\Psi_n-\Psi_n^*(a_+)\Psi_n) dx</math>

⇒ which is proportional to <math> \int_{-\infty}^{\infty} \Psi_n^*\Psi_{n-1} dx - \int_{-\infty}^{\infty}\Psi_n^*\Psi_{n+1} dx = 0 </math>

-which is not a surprising result; for a harmonic oscillator, the average momentum is zero because the particle spends an equal amount of time having positive momentum as an equal but opposite momentum going the other direction.

The expectation of the momentum squared is given by:

<math> <p^2>_\Psi_n = \int_{-\infty}^{\infty}\Psi_n^*(-i sqrt{\frac{m \hbar \omega}{2}} (a_- - a_+))^2\Psi_n dx</math>

<math> = -\frac{m \hbar \omega}{2} \int_{-\infty}^{\infty}\Psi_n^*(a_- - a_+)^2 \Psi_n dx</math>

<math> = -\frac{m \hbar \omega}{2} \int_{-\infty}^{\infty}\Psi_n^*(a_-^2 - a_-a_+ - a_+a_- + a_+^2) \Psi_n dx</math>

-where the first and last terms of the expansion raise and lower the n-value twice, canceling. The middle two cancel because For all Hermitian Operators, their eigenvectors are orthogonal. Here's a link that explains Hermitian Operators in more depth:

http://mathworld.wolfram.com/HermitianOperator.html

(*I'm still a little unsure about this last part, so any clarifications would be appreciated)

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