Quantum Mechanics: Hermitian Operators, Perturbation Theory, and Identical Particles, Lecture notes of Quantum Mechanics

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Quantum Mechanics Problems

David J. Jeffery

Department of Physics & Astronomy

University of Nevada, Las Vegas

Las Vegas, Nevada

Portpentagram Publishing (self-published)

2018 January 1

Introduction

Quantum Mechanics Problems (QMP) is a source book for instructors of introductory quantum mechanics. The book is available in electronic form to instructors by request to the author. It is free courseware and can be freely used and distributed, but not used for commercial purposes. The aim of QMP is to provide digestable problems for quizzes, assignments, and tests for modern students. There is a bit of spoon-finding—nourishing spoon-feeding I hope. The problems are grouped by topics in chapters: see Contents below. The chapter ordering follows roughly the traditional chapter/topic ordering in quantum mechanics textbooks. For each chapter there are two classes of problems: in order of appearance in a chapter they are: (1) multiple- choice problems and (2) full-answer problems. Almost all the problems have complete suggested answers. The answers may be the greatest benefit of QMP. The questions and answers can be posted on the web in pdf format. The problems have been suggested by many sources, but have all been written by me. Given that the ideas for problems are the common coin of the realm, I prefer to call my versions of the problems redactions. Instructors, however, might well wish to find solutions to particular problems from well known texts. Therefore, I give the suggesting source (when there is one and I recall what it was) by a reference code on the extra keyword line: e.g., (Gr-10:1.1) stands for Griffiths, p. 10, problem 1.1. Caveat: my redaction and the suggesting source problem will not in general correspond perfectly or even closely in some cases. The references for the source texts and other references follow the contents. A general citation is usually, e.g., Ar-400 for Arfken, p. 400. At the end of the book are three appendices. The first is set of review problems anent matrices and determinants. The second is an equation sheet suitable to give to students as a test aid and a review sheet. The third is a set of answer tables for multiple choice questions. Quantum Mechanics Problems is a book in progress. There are gaps in the coverage and the ordering of the problems by chapters is not yet final. User instructors can, of course, add and modify as they list. Everything is written in plain TEX in my own idiosyncratic style. The questions are all have codes and keywords for easy selection electronically or by hand. A fortran program for selecting the problems and outputting them in quiz, assignment, and test formats is also available. Note the quiz, etc. creation procedure is a bit klutzy, but it works. User instructors could easily construct their own programs for problem selection. I would like to thank the Department of Physics & Astronomy of the University of Nevada, Las Vegas for its support for this work. Thanks also to the students who helped flight-test the problems at UNLV and other universities.

Contents

1 Classical Physics in Trouble 2 QM Postulates, Schr¨odinger Equation, and the Wave Function 3 Infinite Square Wells and Other Simple Wells 4 The Simple Harmonic Oscillator (SHO) 5 Free Particles and Momemtum Representation 6 Foray into Advanced Classical Mechanics 7 Linear Algebra 8 Operators, Hermitian Operators, Braket Formalism 9 Time Evolution and the Heisenberg Representation 10 Measurement

i

Griffiths, D. J. 1995, Introduction to Quantum Mechanics (Upper Saddle River, New Jersey: Prentice Hall) (Gr) Griffiths, D. J. 2005, Introduction to Quantum Mechanics (Upper Saddle River, New Jersey: Prentice Hall) (Gr2005) Goldstein, H., Poole, C. P., Jr., & Safko, J. L. 2002, Classical Mechanics, 3rd Edition (San Francisco: Addison-Wesley Publishing Company) (GPS) Harrison, W. A. 2000, Applied Quantum Mechanics (Singapore: World Scientific) (Ha) Hodgman, C. D. 1959, CRC Standard Mathematical Tables, 12th Edition (Cleveland, Ohio: Chemical Rubber Publishing Company) (Hod) Jackson, J. D. 1975, Classical Electrodynamics (New York: John Wiley & Sons) Jeffery, D. J. 2001, Mathematical Tables (Port Colborne, Canada: Portpentragam Publishing) (MAT) Keenan, C. W., Wood, J. H., & Kleinfelter, D. C., 1976, General College Chemistry (New York: Harper & Row, Publishers) (Ke) Leighton, R. B. 1959, Principles of Modern Physics (New York: McGraw-Hill Book Company, Inc.) (Le) Mihalas, D. 1978, Stellar Atmospheres, 2nd Edition (San Francisco: W. H. Freeman and Company) (Mi) Morrison, M. A. 1990, Understanding Quantum Mechanics (Upper Saddle River, New Jersey: Prentice Hall) (Mo) Morrison, M. A., Estle, T. L., & Lane, N. F. 1991, Understanding More Quantum Mechanics (Upper Saddle River, New Jersey: Prentice Hall) (MEL) Pathria, R. K. 1980, Statistical Mechanics (Oxford: Pergamon Press) (Pa) Pointon, A. J. 1967, Introduction to Statistical Physics (London: Longman Group Ltd.) (Po) Tipler, P. A. 1978, Modern Physics (New York: Worth Publishers, Inc.) (Ti)

iii

Chapt. 1 Classical Physics in Trouble

Multiple-Choice Problems

001 qmult 00100 1 1 3 easy memory: quantum mechanics

1. The physical theory that deals mainly with microscopic phenomena is:

a) quartz mechanics. b) quarks mechanics. c) quantum mechanics. d) quantum jump mechanics. e) quasi-mechanics.

001 qmult 00200 1 1 1 easy memory: photon energy

2. The photon, the quantum of electromagnetic radiation, has ENERGY:

a) hf = −hω. b) h/λ. c) −hk. d) h^2 f. e) hf 2.

001 qmult 00300 1 1 4 easy memory: photoelectric effect

3. A key piece of evidence for the wave-particle duality of light is: a) the photograph effect. b) the Maxwell’s electrodynamics as summarized in the four Maxwell’s equations. c) the frequency of red light. d) the photoelectric effect. e) the photomagnetic effect.

001 qmult 00400 1 1 1 easy memory: Compton effect

4. Einstein predicted and Compton proved that photons:

a) have linear momentum. b) do not have linear momentum. c) sometimes have linear momentum. d) both have and do not have linear momentum at the same time. e) neither have nor have not linear momentum.

001 qmult 00500 1 4 3 easy deducto-memory: Bohr atom

5. “Let’s play Jeopardy! For $100, the answer is: This model of an atom is of historical and pedagogical interest, but it is of little use in modern practical calculations and from the modern standpoint is probably misleading rather than insight-giving.”

What is , Alex?

1

Chapt. 1 Classical Physics in Trouble 3

001 qmult 00800 1 1 1 easy memory: causality, relativity

8. Einstein ruled out faster than light signaling because: a) it would cause irresolvable causality paradoxes. b) it would not cause irresolvable causality paradoxes. c) it led to irresolvable paradoxes in quantum mechanics. d) it would destroy the universe. e) it had been experimentally verified.

001 qmult 00900 1 1 3 easy memory: EPR paradox

9. The Einstein-Podolsky-Rosen (EPR) paradox was proposed to show that ordinary quantum mechanics implied superluminal signaling and therefore was: a) more or less correct. b) absolutely correct. c) defective. d) wrong in all its predictions. e) never wrong in its predictions.

001 qmult 01000 1 4 3 easy deducto-memory: Bell’s theorem

10. “Let’s play Jeopardy! For $100, the answer is: This theorem (if it is indeed inescapably correct) and the subsequent experiments on the effect the theorem dealt with show that quantum mechanical signaling exceeds the speed of light.” a) What is Dark’s theorem, Alex? b) What is Midnight’s theorem, Alex? c) What is Bell’s theorem, Alex? d) What is Book’s theorem, Alex? e) What is Candle’s theorem, Alex?

Full-Answer Problems

001 qfull 00500 3 5 0 tough thinking: Rutherford’s nucleus Extra keywords: (HRW-977:62P)

1. Rutherford discovered the nucleus in 1911 by bombarding metal foils with alpha particles now known to be helium nuclei (atomic mass 4.0026). An alpha particle has positive charge 2e. He expected the alpha particles to pass right through the foils with only small deviations. Most did, but some scattered off a very large angles. Using a classical particle picture of the alpha particles and the entities they were scattering off of he came to the conclusion that atoms contained most of their mass and positive charge inside a region with a size scale of ∼ 10 −^15 m = 1 fm: this 10−^5 times smaller than the atomic size. (Note fm stands officially for femtometer, but physicists call this unit a fermi.) Rutherford concluded that there must be a dense little core to an atom: the nucleus. a) Why did the alpha particles scatter off the nucleus, but not off the electrons? HINTS: Think dense core and diffuse cloud. What is the force causing the scattering? b) If the alpha particles have kinetic energy 7.5 Mev, what is their de Broglie wavelength?

c) The closest approach of the alpha particles to the nucleus was of order 30 fm. Would the wave nature of the alpha particles have had any effect? Note the wave-particle duality was not even suspected for massive particles in 1911.

4 Chapt. 1 Classical Physics in Trouble

001 qfull 01000 3 5 0 tough thinking: black-body radiation, Wien law Extra keywords: (Le-62) gives a sketch of the derivations

2. Black-body radiation posed a considerable challenge to classical physics which it was partially able to meet. Let’s see how far we can get from a classical, or at least semi-classical, thermodynamic equilibrium analysis. a) Let Uλ be the radiation energy density per wavelength of a thermodynamic equilibrium radiation field trapped in some kind of cavity. The adjective thermodynamic equilibrium implies that the field is homogenous and isotropic. I think Hohlraum was the traditional name for such a cavity. Let’s call the field a photon gas and be done with it—anachronism be darned. Since the radiation field is isotropic, the specific intensity is then given by

B(λ, T ) =

cUλ 4 π

, (Pr.1)

where c is of course the speed of light. Specific intensity is radiation flux per wavelength per solid angle. From special relativity (although there may be some legitimately classical way of getting it), the momentum flux associated with a specific intensity is just B(λ, T )/c. Recall the rest plus kinetic energy of a particle is given by

E =

p^2 c^2 + m^20 c^4 , (Pr.2)

where p is momentum and m 0 is rest mass. From an integral find the expression for the radiation pressure on a specularly reflecting surface:

p =

U , (Pr.3)

where p is now pressure and U is the wavelength-integrated radiation density. Argue that the same pressure applies even if the surface is only partially reflecting or pure blackbody provided the the radiation field and the surface are in thermodynamic equilibrium. HINT: Remember to account for angle of incidence and reflection. b) Now we can utilize a few classical thermodynamic results to show that

U = aT 4 , (Pr.4)

where a is a radiation constant related to the Stefan-Boltzmann constant σ = 5. 67051 × 105 ergs/(cm^2 K^4 ) and T is Kelvin temperature, of course. The relation between a and σ follows from the find the flux leaking out a small hole in the Hohlraum:

F = 2π

0

cU 4 π

μ dμ =

ca 4

T 4 , (P r.5)

where μ is the cosine of the angle to the normal of the surface where the hole is. One sees that σ = ca/4. Classically a cannot be calculated theoretically; in quantum mechanical statistical mechanics a can be derived. The proportionality U ∝ T 4 can, however, be derived classically. Recall the 1st law of thermodynamics:

dE = T dS − p dV , (Pr.6)

where E is internal energy, S is entropy, and V is volume. Note that ( ∂E ∂S

V

= T and

∂E

∂V

S

= −p , (Pr.7)

6 Chapt. 1 Classical Physics in Trouble

where k 1 and k 2 had to be determined from the fit. Wien’s law works well for short wavelengths (x (^) ∼< xmax), but gives a poorish fit to the long wavelengths (x (^) ∼> xmax) (Pa-190, but note the x there is the inverse of the x here aside from a constant). The Rayleigh-Jeans law derived from a rather different classical starting picture gave a good fit to long wavelengths (x >> xmax), but failed badly at shorter wavelengths (Pa-190, but note the x there is the inverse of the x here aside from a constant). In fact the Rayleigh-Jeans law goes to inifinity as x goes to zero and the total energy in a Rayleigh-Jeans radiation field is infinite (Le-64): this is sometimes called the ultraviolet catastrophe (BFG-106). The correct black-body specific intensity law was derived from a primitive quantum theory by Max Planck in 1900 (BFG-106). Planck obtained an empirically excellent fit to the black-body specific intensity and then was able to derive it from his quantum hypothesis. The Rayleigh-Jeans and Planck laws are the subject for another question.

001 qfull 01100 2 5 0 moderate thinking: Bohr atom

3. In 1913, Niels Bohr presented his model of the hydrogen atom which was quickly generalized to the hydrogenic atom (i.e., the one-electron atom of any nuclear charge Z). This model correctly gives the main hydrogenic atom energy levels and consists of a mixture of quantum mechanical and classical ideas. It is historically important for showing that quantization is somehow important in atomic structure and pedagogically it is of interest since it shows how simple theorizing can be done. But the model is, in fact, incorrect and from the modern perspective probably even misleading about the quantum mechanical nature of the atom. It is partially an accident of nature that it exists to be found. Only partially an accident since it does contain correct ingredients. And it is no accident that Bohr found it. Bohr knew what he wanted: a model that would successfully predict the hydrogen atom spectrum which is a line spectrum showing emission at fixed frequencies. He knew from Einstein’s photoelectric effect theory that electromagnetic radiation energy was quantized in amounts hν where h = 6.62606896(33) × 10 −^27 erg s was Planck’s constant (which was introduced along with the quantization notion to explain black- body radiation in 1900) and ν was frequency of the quantum of radiation. He recognized that Planck’s constant had units of angular momentum. He knew from Rutherford’s nuclear model of the atom that the positive charge of an atom was concentrated in region that was much smaller than the atom size and that almost all the mass of the atom was in the nucleus. He knew that there were negative electrons in atoms and they were much less massive than the nucleus. He knew the structure of atoms was stable somehow. By a judicious mixture of classical electromagnetism, classical dynamics, and quantum ideas he found his model. A more sophisticated mixture of these concepts would lead to modern quantum mechanics. Let’s see if we can follow the steps of the ideal Bohr—not the Bohr of history. NOTE: This a semi-classical question: Bohr, ideal or otherwise, knew nothing of the Schr¨odinger equation in 1913. Also note that this question uses Gaussian CGS units not MKS units. The most relevant distinction is that electric charge

eCGS =

eMKS √ 4 πǫ 0

which implies the fine structure constant in CGS is

α =

e^2 − hc

Astronomy is all Gaussian CGS by the way. a) Bohr thought to build the electron system about the nucleus based on the electrostatic inverse square law with the electron system supported against collapse onto the nucleus by angular momentum. The nucleus was known to be much tinnier than the electron system

Chapt. 1 Classical Physics in Trouble 7

which gives the atom its volume. The nucleus could thus be a considered an immobile point center of force at the origin of the relative nucleus-electron coordinate system frame. This frame is non-inertial, but classically can be given an inertial-frame treatment if the electron is given a reduced mass given by

m =

memnucleus me + mnucleus ≈ me

me mnucleus

where me the electron mass and mnucleus is the nucleus mass. The approximation is valid for me/mnucleus << 1 which is true of hydrogen and most hydrogenic systems, but not, for example, for positronium (a bound electron and positron). The electron—there is only one in a hydrogenic atom—was taken to be in orbit about the nucleus. Circular orbits seemed the simplest way to proceed. The electrostatic force law (in Gaussian cgs units) in scalar form for a circular orbit is

F^ ~ = − Ze

2 r^2

r ,ˆ

where Ze is the nuclear charge, e is the electron charge, and r is the radial distance to the electron, and ˆr is a unit vector in the radial direction. What is the potential energy of the electron with the zero of potential energy for the electron at infinity as usual? HINT: If the result isn’t obvious, you can get it using the work-potential energy formula:

V = −

F^ ~ · d~r + constant.

b) Using the centripetal force law (which is really F = ma for uniform circular motion)

F^ ~ = − mv

2 r r ,ˆ

find an expression for the classical kinetic energy T of the electron in terms of Z, e, and r alone. c) What is the total energy of the electron in the orbit?

d) Classically an accelerating charge radiates. This seemed well established experimentally in Bohr’s time. But an orbiting electron is accelerating, and so should lose energy continuously until it collapses into the nucleus: this catastrophe obviously doesn’t happen. Electrons do not collapse into the nucleus. Also they radiate only at fixed frequencies which means fixed quantum energies by Einstein’s photoelectric effect theory. So Bohr postulated that the electron could only be in certain orbits which he called stationary states and that the electron in a stationary state did not radiate. Only on transitions between stationary states (sometimes called quantum jumps or leaps) was there an emission of radiation in a quantum of radiation or (to use an anachronism) a photon. To get the fixed energies of emission only certain energies were allowed for the stationary states. But the emitted photons didn’t come out with equally spaced energies: ergo the orbits couldn’t be equally spaced in energy. From the fact that Planck’s constant h has units of angular momentum, Bohr hypothesized the orbits were quantized in equally spaced amounts of angular momentum. But h was not the spacing that worked. Probably after a bit of fooling around, Bohr found that h/(2π) or, as we now write it, −h^ was the spacing that gave the right answer. The allowed angular momenta were given by L = n h−^ ,

Chapt. 1 Classical Physics in Trouble 9

d) Sketch the behavior of ∆λ as a function of θ. What is the shift formula in the non-relativistic limit: i.e., when λ → ∞.

001 qfull 00150 3 5 0 tough thinking: Einstein, Runyon Extra keywords: Bosher

5. “God does not play dice”—Einstein. Discuss.

Chapt. 2 QM Postulates, Schr¨odinger Equation, and the Wave Function

Multiple-Choice Problems

002 qmult 00080 1 1 2 easy memory: wave-particle duality

1. The nebulous (and sometimes disparaged) concept that all microscopic physical entities have both wave and particle properties is called the wave-particle: a) singularity. b) duality. c) triality. d) infinality. e) nullility.

002 qmult 00090 1 4 5 easy deducto-memory: Sch eqn

2. “Let’s play Jeopardy! For $100, the answer is: The equation that governs (or equations that govern) the time evolution of quantum mechanical systems in the non-relativistic approximation.” What is/are , Alex?

a) F~net = m~a b) Maxwell’s equations c) Einstein’s field equations of general relativity d) Dirac’s equation e) Schr¨odinger’s equation

002 qmult 00100 1 1 1 easy memory: Sch eqn compact form

3. The full Schr¨odinger’s equation in compact form is:

a) HΨ = i h−^

∂t

. b) HΨ = −h^

∂t

. c) HΨ = i

∂t

. d) HΨ = i h−^

∂x

e) H−^1 Ψ = i h−^

∂t

002 qmult 00110 1 1 3 easy memory: Hamiltonian operator

4. The energy operator in quantum mechanics,

H = −

−h 2

2 m

∂^2

∂x^2

• V (x)

(here given for 1 particle in one dimension) is called the: a) Lagrangian b) Laplacian c) Hamiltonian d) Georgian e) Torontonian

002 qmult 00200 1 4 3 easy deducto-memory: Born postulate Extra keywords: mathematical physics

5. “Let’s play Jeopardy! For $100, the answer is: The postulate that the wave function Ψ(~r ) is quantum mechanics is a probability amplitude and |Ψ(~r )|^2 is a probability density for localizing a particle at ~r on a ‘measurement’.” What is , Alex? a) Schr¨odinger’s idea b) Einstein’s notion c) Born’s postulate d) Dirac’s hypothesis e) Death’s conclusion

002 qmult 00210 1 1 1 easy memory: QM probability density

6. In the probabilistic interpretation of wave function Ψ, the quantity |Ψ|^2 is:

10

12 Chapt. 2 QM Postulates, Schr¨odinger Equation, and the Wave Function

a) fudge factor. b) dynamical variable. c) universal constant. d) constant of the motion. e) constant of the stagnation.

002 qmult 00520 1 4 5 easy deducto-memory: Ehrenfest’s theorem

15. Ehrenfest’s theorem partially shows the connection between quantum mechanics and: a) photonics. b) electronics. c) special relativity. d) general relativity. e) classical mechanics.

002 qmult 00600 1 4 5 easy deducto-memory: uncertainty principle 1

16. “Let’s play Jeopardy! For $100, the answer is: It describes a fundamental limitation on the accuracy with which we can know position and momentum simultaneously.” What is , Alex? a) Tarkovsky’s doubtful thesis b) Rublev’s ambiguous postulate c) Kelvin’s nebulous zeroth law d) Schr¨odinger’s wild hypothesis e) Heisenberg’s uncertainty principle

002 qmult 00610 1 4 5 easy deducto-memory: uncertainty principle 2

17. “Let’s play Jeopardy! For $100, the answer is: ∆x∆p ≥ −h/2 or σxσp ≥ −h/2. What is , Alex?

a) an equality b) a standard deviation c) the Heisenberg CERTAINTY principle d) the Cosmological principle e) the Heisenberg UNCERTAINTY principle

002 qmult 00700 1 1 4 easy memory: Schr. eqn. separation of variables

18. The time-independent Schr¨odinger equation is obtained from the full Schr¨odinger equation by: a) colloquialism. b) solution for eigenfunctions. c) separation of the x and y variables. d) separation of the space and time variables. e) expansion.

002 qmult 00720 1 1 1 easy memory: stationary state

19. A system in a stationary state will: a) not evolve in time. b) evolve in time. c) both evolve and not evolve in time. d) occasionally evolve in time. e) violate the Heisenberg uncertainty principle.

002 qmult 00800 1 4 2 easy deducto-memory: orthogonality property

20. For a Hermitian operator eigenproblem, one can always find (subject to some qualitifications perhaps—but which are just mathemtical hemming and hawwing) a complete set (or basis) of eigenfunctions that are:

a) independent of the x-coordinate. b) orthonormal. c) collinear. d) pathological. e) righteous.

002 qmult 00810 1 4 2 easy deducto-memory: basis expansion Extra keywords: mathematical physics

21. “Let’s play Jeopardy! For $100, the answer is: If it shares the same same range as a basis set of functions and is at least piecewise continuous, then it can be expanded in the basis with a vanishing limit of the mean square error between it and the expansion.” What is a/an , Alex? a) equation b) function c) triangle d) deduction e) tax deduction

002 qmult 00820 1 4 5 easy deducto-memory: general Born postulate Extra keywords: mathematical physics

Chapt. 2 QM Postulates, Schr¨odinger Equation, and the Wave Function 13

22. “Let’s play Jeopardy! For $100, the answer is: The postulate that expansion coefficients of a wave function in the eigenstates of an observable are the probability amplitudes for wave function collapse to eigenstates of that observable.” What is , Alex? a) the special Born postulate b) the very special Born postulate c) normalizability d) the mass-energy equivalence e) the general Born postulate

002 qmult 00830 1 1 4 easy memory: basis expansion physics

23. The expansion of a wave function in an observable’s basis (or complete set of eigenstates) is a) just a mathematical decomposition. b) useless in quantum mechanics. c) irrelevant in quantum mechanics. d) not just a mathematical decomposition since the expansion coefficients are probability amplitudes. e) just.

020 qmult 00840 1 4 5 easy deducto-memory: wave function collapse Extra keywords: mathematical physics

24. “Let’s play Jeopardy! For $100, the answer is: It is a process in quantum mechanics that some decline to mention, some believe to be unspeakable, some believe does not exist (though they have got some explaining to do about how one ever measures anything), some believe should not exist, and that some call the fundamental perturbation (but just once per textbook).” What is , Alex? a) the Holy b) the Unholy c) the Unnameable d) the 4th secret of the inner circle e) wave function collapse

002 qmult 00900 1 4 1 easy deducto-memory: macro object in stationary state

25. “Let’s play Jeopardy! For $100, the answer is: A state that no macroscopic system can be in except arguably for states of Bose-Einstein condensates, superconductors, superfluids and maybe others sort of.” What is a/an , Alex? a) stationary state b) accelerating state c) state of the Union d) state of being e) state of mind

002 qmult 01000 1 1 5 easy memory: stationary state is radical

26. A stationary state is: a) just a special kind of classical state. b) more or less a kind of classical state. c) voluntarily a classical state. d) was originally not a classical state, but grew into one. e) radically unlike a classical state.

002 qmult 01100 1 1 4 easy memory: macro system in a stationary state

27. Except arguably for certain special cases (superconductors, superfluids, and Bose-Einstein condensates), no macroscopic system can be in a: a) mixed state. b) vastly mixed state. c) classical state. d) stationary state. e) state of the union.

002 qmult 01200 1 1 2 easy memory: transitions

28. Transitions between atomic or molecular stationary states (sometimes, but actually rarely, called quantum jumps) are: a) only collisional. b) both collisional and radiative. c) only radiative.

Chapt. 2 QM Postulates, Schr¨odinger Equation, and the Wave Function 15

002 qfull 00200 2 3 0 moderate math: probability needle 1 Extra keywords: (Gr-10:1.3) probability and continuous variables

3. An indicator needle on a semi-circular scale (e.g., like a needle on car speedometer) bounces around and comes to rest with equal probability at any angle θ in the interval [0, π]. a) Give the probability density ρ(θ) and sketch a plot of it. b) Compute the 1st and 2nd moments of the distribution (i.e., 〈θ〉 and

θ^2

) and the variance and standard deviation. c) Compute 〈sin θ〉, 〈cos θ〉, 〈sin^2 θ〉, and 〈cos^2 θ〉.

002 qfull 00210 3 5 0 tough thinking: 2-variable probability density Extra keywords: (Gr-11:1.5) dropping a needle on lines needle 2

4. Nun f¨ur eine kleine teufelische problem. Say you drop at random with equal likelihood of landing in any orientation and location a needle of length ℓ onto a sheet of paper with parallel lines a distance ℓ apart. What is the probability of the needle crossing (or at least touching) a line? Let’s be nice this time and break it down. a) Mentally mark one end of needle red. Then note that really we only need to consider one band on the paper between two parallel lines and the case where the red end lies between them as a given. Why is this so? b) So now we consider that the red end lands in one band at a point x between −ℓ/2 and ℓ/2. Note we put the origin at the center since almost always one ought to exploit symmetry. What is the probability density for the red end to land anywhere in the band? What is the probability density for the needle for the orientation of the needle in θ measured from the x-axis? Why do you only need to consider θ ∈ [0, π]? c) Now we don’t care about the orientation itself really: we just care about it’s projection on the x-axis. Call that projection x′. What is the probability density for x′? What is the range of x′^ allowed? HINT: The probability of landing in dθ and a corresponding dx′ must be equal.

d) The joint probability density for x and x′^ is

ρ(x)ρ(x′).

You now have to integrate up all the probability for x′^ + x ≥ ℓ/2 and for x′^ + x ≤ −ℓ/ 2 and sum those two probabilities. The sum is the solution probability of course.

002 qfull 00220 1 3 0 easy math: Gaussian probability density Extra keywords: (Gr-11:1.6)

5. Consider the Gaussian probability density

ρ(x) = Ae−λ(x−a)

2 ,

where A, a, and λ are constants. a) Determine the normalization constant A. b) The nth moment of a probability density is defined by

〈xn〉 =

−∞

xnρ(x) dx.

Determine the 0th, 1st, and 2nd moments of the Gaussian probability density.

16 Chapt. 2 QM Postulates, Schr¨odinger Equation, and the Wave Function

c) For the Gaussian probability density determine the mean, mode, mediam, variance σ^2 , and standard deviation (or dispersion) σ. d) Sketch the Gaussian probability density.

002 qfull 00300 2 3 0 moderate math: analyzing a triangular hat wave function Extra keywords: (Gr-13:1.7)

6. At some time a triangular hat wave function is given by

Ψ(x, t) =

A

x a

, x ∈ [0, a];

A

b − x b − a

, x ∈ [a, b];

0 otherwise,

where A, a, and b are constants. a) Sketch Ψ and locate most probable location for a particle (i.e., the mode of the |Ψ|^2 probability distribution). b) Determine the normalization constant A in terms of a and b. Recall the difference between wave function and probability distribution here and in the later parts of this question. c) What are the probabilities of being found left and right of a, respectively? d) What is 〈x〉?

002 qfull 00310 2 5 0 moderate thinking: probability conservation Extra keywords: (Gr-13:1.9) probability current

7. The expression for the probability that a particle is in the region [−∞, x] (i.e., the cumulative probability distribution function) is

P (x, t) =

∫ (^) x

−∞

|Ψ(x′, t)|^2 dx′^.

a) Find an explicit, non-integral formula for ∂P (x, t)/∂t given that the wave function is normalizable at time t. Simplify the formula as much as reasonably possible. HINT: Make use of the physics: i.e., the Schr¨odinger equation itself. This is a common trick in quantum mechanics and, mutatis mutandis, throughout physics. It probably helps to let the dummy variable in the integral be x and the endpoint a while doing the math. b) Recall momentum observable is

pop =

−h i

∂x

Substitute pop into the formula derived in part (a) and simplify as much as possible. In the simplification, make use of the real-part function Re which has the property that

Re(z)

is the real part of complex variable z. For example, if z = x + iy, then

Re(z) = Re(x + iy) = x.

HINT: Note that −popΨ∗^ = (popΨ)∗^.