Download Problem Set 2 for OPTI 511, Spring 2009: Operators and Expectation Values and more Assignments Chemistry in PDF only on Docsity!
OPTI 511, Spring 2009 Problem Set 2 Prof. R.J. Jones
Due Feb. 5
Operators
Simply put, an operator is an instruction to do something on the function that follows; that something may be multiplication or differentiation, or combinations thereof. Individual operators can also be multiplied (oreder matters!) or added to produce new operators. Operators also often (but don’t al- ways) represent physically measurable quantities, or “observables”. For example, the spatial position of an object is a physically measurable quantity. It is thus called an “observable.” In quantum mechanics, any quantity of interest Q will be represented by an operator, denoted Qˆ, whether or not Q is an observable. Position, momentum, energy, and angular momentum are a few examples of observables, and each of these quantities has an associated operator. Operators can also represent non-physical actions or quantities. For example, we can define an operator Cˆ that multiplies a function by i times the fifth root of position x and adds the result to the 7th^ derivative of the function (with respect to position) times a constant B:
Cˆ = i √^5 x + B · d
7 dx^7
This is a nonsensical operator, but we could nevertheless evaluate Cˆ acting on a wavefunction Ψ(x, t). We could also evaluate the expectation value 〈C〉 using the method defined in class. In this case, it would be hard to justify why we’d care about the expectation value of this silly quantity, or why we’d even want to use the term expectation value when there is apparently no measurement that relates to Cˆ, but the formalism has been defined to be general enough to handle such cases. To make matters even more confusing, the form of an operator (such as the operator ˆpx represent- ing momentum along the x direction) depends on the functional space of the function or wavefunction following it. For example, the operator for momentum along the x direction is ˆpx = −i¯h (^) ∂x∂ , but only if we are working with functions of position x. If we are, the functional space is position or coordinate space. However, if the functional space of our wavefunction is momentum space, meaning that the wavefunction is expressed as a function of momentum px, the momentum operator is simply ˆpx = px. Likewise, the position operator is simply ˆx = x in the coordinate space representation, but ˆx = i¯h (^) ∂p∂x in momentum space. If this sounds confusing, think of it like this: classically, the position of a particle versus time can be labeled x(t), and the momentum is m dxdt. This would be a coordinate space repre- sentation. Or, we could label the momentum as p(t) and the position as (1/m)
∫ p(t)dt. This would be a momentum-space representation. There’s a bit more to it in QM, but the point is that a physical quantity can be expressed in multiple ways depending on what is known. In QM, the way to express a particular quantity depends on which functional space is used; it’s usually important is to stick with a single functional space (such as momentum or position) during mathematical manipulation.
- Math exercise For the following problems, an operator Qˆ and a function F (not necessarily a wavefunction) are given. In parts (a) to (e), evaluate QFˆ , and simplify as much as possible.
Example: Qˆ(x) = x^2 + B (^) dxd , and F (x) = A exp[−x^3 /a^3 ]. Treat A, B and a as constants.
Solution:
QFˆ (x) = x^2 A exp[−x^3 /a^3 ] + B · A · (− 3 x^2 /a^3 ) exp[−x^3 /a^3 ] = (x^2 − 3 Bx^2 /a^3 )A exp[−x^3 /a^3 ] = x^2 (1 − 3 B/a^3 )F (x) (the answer).
(a) Qˆ(x) = −i¯h (^) ∂x∂ , (the 1-D momentum operator in coordinate space), and F (x, t) = Aei(kx−ωt).
(b) Qˆ(x) = − ¯h
2 2 m
∂^2 ∂x^2 , (the 1-D kinetic energy operator in coordinate space), and^ F^ (x, t) =^ Ae
i(kx−ωt).
(c) Qˆ(x) = ˆxpˆx = −i¯hx (^) ∂x∂ , (a possibly meaningless operator formed out of the product of the position and momentum operators), and F (x) = Ae−x (^2) /(2a (^2) ) .
(d) Qˆ(x, y, z) = −i¯h(x (^) ∂y∂ − y (^) ∂x∂ ), (an angular momentum operator in 3-D coordinate space), and
F (x, y, z) = Ae−r(x,y,z)/(2a)^ where r(x, y, z) =
√ x^2 + y^2 + z^2 and r is always positive.
(e) Qˆ(t) = i¯h (^) ∂t∂ , (an energy operator), and F (t) = Ae−x^2 /a^2 e−iωt.
- Wavefunction algebra
Suppose that a particle in a one-dimensional potential well is represented by the spatial wavefunction
ψ(x) = A · (x/a) · e−x (^2) /(2a (^2) ) ,
where A and a are constants. x and a have units of length.
(a) What are the units of A?
(b) Normalize ψ(x) (that is, find A in terms of a). Remember that x extends from −∞ to +∞.
(c) Sketch ψ(x) and |ψ(x)|^2 (on two separate plots) as a function of x. Be sure to label your axes!
(d) Calculate the expectation values of x, x^2 , px, and p^2 x in terms of a [be sure to use the value of A that you found in part (b)]. Recall that in coordinate space (as opposed to momentum space), the operator for x-direction momentum px is ˆpx = −i¯h (^) ∂x∂ , and the operator for p^2 x is thus −¯h^2 ∂ 2 ∂x^2. Remember: the placement of an operator within an expectation value integral is in general very im- portant; use the formal method for determining expectation values that was discussed in class.
(e) Using standard deviations to define the widths ∆x and ∆px of the wavefunction (i.e., the un- certainty in position and momentum), what is the product ∆x · ∆px? Is this product consistent with the uncertainty principle?
Things to note in this problem: (1) The wavefunction here is an eigenfunction of the Time-Independent Schr¨odinger Equation for a particle in a one-dimensional harmonic oscillator potential, as you will see in problem 4 below. (We will soon discuss what this means in class). The eigenfunctions have a Hermite-Gaussian functional form like the transverse modes of an optical resonator. (2) 〈x〉 does not necessarily correspond to a location where the particle might actually be found, i.e., where the wavefunction has a non-zero amplitude. Such is the case in the above problem. If this troubles you, recall that the “expectation value” for the number of dots obtained in a roll of a standard 6-sided die is 3.5, even though this value can not be obtained in a single roll. (3) This problem illustrates that momentum information can be determined directly from the spatial representation of the wavefunction; using the momentum representation of the wavefunction is not necessary here, but in general it might make a particular problem easier to solve or understand.
equation is satisfied only for particular eigenfunctions ψn(x), each with an associated eigenvalue En that necessarily has units of energy. (The subscript n is often used to characterize one of the many eigenfunctions of a given operator Hˆ.) A common goal in quantum mechanics is to determine the eigenvalues and eigenfunctions associated with specific operators.
In our 1-D problem, we define Hˆ(x) as
Hˆ = − ¯h
2 2 m
d^2 dx^2
The relevance of the constants m and ω will be clarified later in the problem. Your task is to verify that ψn(x) = Anxne−x
(^2) /(2a (^2) )
is a solution of the eigenvalue equation only for particular values of a and the integer n. You will see that there are only a couple of values of n for which this particular form of ψn is a solution. For example, n = 6 and n = 122 do not produce eigenfunctions of the operator Hˆ as defined above. In finding which ψn are eigenfunctions, you will also be determining the eigenvalues En associated with each of the eigenfunctions found. An is a normalization coefficient; treat it as a constant rather than a function of x.
Solve this problem by the following steps:
(a) Evaluate Hψˆ n(x), using the above expressions for Hˆ and ψn(x).
(b) Set your answer from (a) equal to the product Enψn(x), and simplify the expression as much as you can as you solve for En. En will be a constant and independent of x only if n and a satisfy stringent conditions. What are the two values of n and the value for a (in terms of the other constants ¯h, m, and ω) such that En can be made a constant?
(c) What are these two values of En, in terms of ¯h and ω only? You will need to substitute in the value you found for a. Does your answer have units of energy? It will if ω is an angular frequency.
(d) Armed with your answers from above, assume that ψn(x) is a wavefunction that represents a particle, and normalize it for the two different values of n (that is, solve for An for the two cases).
(e) On separate plots, sketch the two eigenfunctions that you found, and label each sketch with the associated value of n (ie, n =?).
(f) Compare the two eigenfunctions with the function in problem 2, and look for any similarities.
While you haven’t fully solved the Schr¨odinger equation for the harmonic oscillator, what you have done in this problem is to verify the correct expressions for two eigenfunctions of the harmonic oscil- lator potential V (x) = (1/2)mω^2 x^2. Here, m is the mass of a particle in a harmonic potential well of oscillation frequency ω. This type of potential well describes the potential energy versus position of a mass on a spring with spring constant mω^2 , as well as numerous other problems. You have also determined the eigenenergies En associated with two different eigenfunctions.
There are actually an infinite number of eigenfunctions (or modes) to the harmonic oscillator po- tential. Be sure to realize that only two have the functional form used in this problem! What you found here was that none of the others can be expressed in the form given above. As it turns out, the set of eigenfunctions have Hermite-Gaussian shapes, like the modes of an optical resonator.
5. EXTRA CREDIT
This is a problem that is intended to help you build insight into Hermite polynomials, Hermite- Gaussian functions, superpositions of wavefunctions, and the harmonic oscillator problem mentioned above (although you may need to wait a bit longer to see the specific relevance). In this problem, you are to write computer code to construct and plot selected Hermite-Gaussian functions, and selected superpositions. Matlab is an excellent software package to use for this problem. Other programming and graphing packages may work well too. (A solution will be provided in Matlab.) For credit on this problem, you will need to turn in a copy of your computer code as well as the results.
Mathematical and Physical foundation for this problem
The normalized eigenfunctions of the operator Hˆ given in problem 4 take the following form:
ψn(x) = (πa^2 )−^1 /^4 · (2nn!)−^1 /^2 · Hn( xa ) · exp (− 12 (x/a)^2 ).
In this expression, n is any integer greater than or equal to zero, a and x are lengths (just like in problem 4 above), and Hn( xa ) is the Hermite polynomial of order n (and is not to be confused with the operator Hˆ). Programming will be simpler if we make a change of variables by re-scaling all lengths in the function ψn by the length a. In other words, we will introduce a new variable u defined as u ≡ x/a. Since all lengths will be in terms of length scale a, including a itself, the coefficient in front of Hn will also change and become π−^1 /^4 · (2nn!)−^1 /^2.
To clearly indicate that we’re working in a re-scaled coordinate system (that is now free of dimensional units), call the newly re-scaled function Fn(u) instead of ψn(x):
Fn(u) = π−^1 /^4 · (2nn!)−^1 /^2 · Hn(u) · exp (−u^2 /2).
The first two Hermite polynomials are:
H 0 (u) = 1 H 1 (u) = 2u
The following relation allows all higher-order Hermite polynomials to then be generated:
Hn(u) = 2uHn− 1 (u) − 2(n − 1)Hn− 2 (u).
(a) Consider the following way of writing a Hermite polynomial: Hn(u) =
∑j=n j=0 cnu n. Begin your
computer program by writing code that determines the coefficients cn of all Hermite polynomials up to order N = 15. For example, and to check your code, the Hermite polynomial of order n = 4 is H 4 (u) = 12 − 48 u^2 + 16u^4 , for which c 0 = 12, c 2 = −48, c 4 = 16, and all other cn values are zero.
You may look for the answers in a book, or in some computer packages, and skip this part of the problem. But for credit on this part of the problem you will need to determine them yourself and turn in a printed copy of your code that generates the coefficients. You do not need to turn in a list of all of the coefficients! Be careful with the indices n. For instance, Matlab’s vectors and matrices must be referenced by indices starting with 1, while the Hermite polynomials start with index n = 0. So you should try to make sure you do not confuse yourself on this!
