OPTI 511R, Spring 2018 Problem Set 1 Prof. R.J. Jones

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1 OPTI 511R, Spring 2018 Problem Set 1 Prof. R.J. Jones Due in class Thursday, Jan 18, 2018 Math Refresher The first 11 questions are just to get you to review some very basic mathematical concepts needed for this course that I expect to be already familiar to you. (For these first 11 questions you may just want to provide your answers in the space provided.) Dimensional units 1. If f(x) 2 dx = 1, and if x has dimensional units of length, what are the dimensional units of f(x)? Differentiation 2. Let F (x, t) e x2 /a 2 e iωt, where x and t are independent variables with units of length and time (respectively), and a and ω are constants. (a.) What are the dimensional units of a? (b.) Evaluate df /dt. (c.) Evaluate df /dx. 1

2 Integration 3. Given that e u2 du = π, solve the following integral: e x2 /a 2 dx. Complex Numbers. Suppose that the complex number z can be written as either z = x + iy or z = Ae iθ, with x, y, A, and θ all real numbers. (a.) Write A in terms of x and y. (b.) Write θ in terms of x and y (there are various correct expressions). Differential Equations y(x) = m y(x). Give a function y(x) that solves this equation. 5. d dx 6. d 2 dx 2 y(x) = m 2 y(x). Give a function y(x) that solves this equation. 2

3 Linear Algebra 7. Suppose M List the eigenvalues of M. 8. Suppose v 2. Evaluate v. 9. Suppose M and v 0 1. Evaluate M v. 3

4 Interpretation and sketching 10. Let y(x) = e x2 /a 2. Sketch this function, including labeled Cartesian axes. Indicate the points x = a and x = a. 11. A section of the curve y = x 2 /a is plotted in the following figure. If x has dimensional units of length, and a = meters and b = 2 meters, give a value (with dimensional units) for the area of the shaded region.

5 12. Clasical Bohr orbital period Using the simple mdodel for the Bohr atom discussed in class, calculate the time it takes for the electron to orbit the nucleus in the hydrogen ground state. This is sometimes referred to as the natural timescale for electron dynamics in atomic systems. 13. debroglie wavelength The debroglie wavelength associated with a particle of mass m moving with velocity v, and therefore momentum p = mv, is λ db = 2π h/p. As you can see, the debroglie wavelength depends upon mass and velocity. We can thus make the generalization that quantum mechanical effects, in particular the wave nature of matter, become large enough to be measurable for very slow and small mass particles. As a general idea, the wave nature of matter needs to be considered when the debroglie wavelength becomes larger than the particle size, or when it is comparable to the spacing between particles. (Remember: the size of a particle is completely independent of the debroglie wavelength of that particle.) In this problem, you will calculate the debroglie wavelength associated with massive particles at different temperatures, expressed in units of absolute temperature (the Kelvin scale, where 0 K is absolute zero, and room temperature is 300 K). To find the particle velocity, use the 1-D relation k B T = mv 2 between temperature T and kinetic energy, where k B is Boltzmann s constant k B = J/K. Calculate λ db for the following: (a) a hydrogen atom of mass m = kg at room temperature. Compare this answer with the size of a hydrogen atom, about m (this is roughly the atom s diameter). (b) a He atom, of mass m = kg, at a temperature of T = 2 K, the temperature range for superfluid effects to occur in liquid helium. (c) a 87 Rb atom, of mass m = kg, at room temperature. (d) a 87 Rb atom at a temperature of T = 10 8 K, a remarkable but achievable temperature for some current physics experiments in the field of Bose-Einstein condensation. (e) an electron of mass kg and a velocity of m/s, a number we might associate with the electron s velocity in an atom. How does this answer compare with the size of a hydrogen atom [see part (a)]? (f) Calculate the debroglie wavelength for a human-sized mass moving at a walking speed. 5

6 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 (oredr matters!) or added to produce new operators. Operators also often (but don t always) 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 ˆQ, 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 Ĉ that multiplies a function by i times the fifth root of position x and adds the result to the 7 th derivative of the function (with respect to position) times a constant B: Ĉ = i 5 x + B d 7 dx 7. (1) This is a nonsensical operator, but we could nevertheless evaluate Ĉ 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 Ĉ, 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 ˆp x representing 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 ˆp x = 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 p x, the momentum operator is simply ˆp x = p x. 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 dx dt. This would be a coordinate space representation. 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. 6

7 1. Math exercise For the following problems, an operator ˆQ and a function F (not necessarily a wavefunction) are given. In parts (a) to (e), evaluate ˆQF, and simplify as much as possible. Example: ˆQ(x) = x 2 + B d dx, and F (x) = A exp[ x3 /a 3 ]. Treat A, B and a as constants. Solution: ˆQF (x) = x 2 A exp[ x 3 /a 3 ] + B A ( 3x 2 /a 3 ) exp[ x 3 /a 3 ] = (x 2 3Bx 2 /a 3 )A exp[ x 3 /a 3 ] = x 2 (1 3B/a 3 )F (x) (the answer). (a) ˆQ(x) = i h x, (the 1-D momentum operator in coordinate space), and F (x, t) = Aei(kx ωt). 2 (b) ˆQ(x) = h2 2m, (the 1-D kinetic energy operator in coordinate space), and F (x, t) = Ae x i(kx ωt). 2 (c) ˆQ(x) = ˆxˆp x = i hx x, (a possibly meaningless operator formed out of the product of the position and momentum operators), and F (x) = Ae x2 /(2a 2). (d) ˆQ(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) ˆQ(t) = i h t, (an energy operator), and F (t) = Ae x2 /a 2 e iωt. 7

The Schroedinger equation

The Schroedinger equation After Planck, Einstein, Bohr, de Broglie, and many others (but before Born), the time was ripe for a complete theory that could be applied to any problem involving nano-scale