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Physics 264L: What to Know and Some Review Questions for Midterm 2 What you should know The following are most of the key facts, concepts, and skills that you should know. 1. You should continue to know by memory the values of the fundamental constants ℏ, k, c, e, K, G, me , and mp to the nearest power of ten in SI units, and you should be able to carry out order-of-magnitude estimates to the nearest power of ten quickly and without having to multiply or divide any digits. 2. You should understand and be able to explain in one to two paragraphs the key details of four experiments that established the particle nature of the photon and the wave nature of massive particles: (a) the photoelectric eﬀect which confirmed that E = hf . (b) the Compton eﬀect that established that photons act like point particles. (c) the Davisson-Germer experiment, which was the first to confirm that electrons have wave-like properties and the wavelength was precisely in accord with the de Broglie relation λ = p/h. (d) the two-slit interference experiment carried out with a monoenergetic beam of electrons, which established both the wave-propert of electrons according to λ = p/h and demonstrated directly the probabilistic meaning of the wave function Ψ. 3. How to normalize a given function f (x) so that ⟨f |f ⟩ = 1. 4. How to determine if two wave functions are orthogonal to one another. 5. You should know how to calculate the phase speed vϕ and group speed vg of waves satisfying some given dispersion relation ω = ω(k), and you should know how to derive heuristically a linear wave equation consistent with a given dispersion relation. 6. For a piecewise constant potential V (x) that is finite everywhere, you should know how to use continuity of the wave function ψ(x) and continuity of its spatial derivative dψ/dx to obtain equations that let you solve the discrete bound-state energies En and calculate explicit wave function solutions for those energies. 7. For a potential V (x) containing one or more Dirac delta functions, you should know how to use continuity of the wave function and integration over small spatial regions centered on each delta function to obtain equations that allow one to solve for the bound-state energy values. 8. You should know how to determine visually and qualitatively if some given transcendental equation has one or more solutions. (A conceptual example: what is the maximum number of intersections a circle can have with the sine function? 9. You need to know how to prove key results involving solutions of the one-dimensional Schrodinger equations such as: (a) the orthogonality of bound states belonging to diﬀerent energy values. (b) that the 1d time-dependent Schrodinger equation preserves probabilities (the area under the probability density curve |Ψ|2 is always equal to 1 if it starts oﬀ being equal to 1. (c) relations like ⟨p⟩ = md⟨x⟩/dt. 1 All of these results involve using the definition of the integral dot product, the fact that a wave function Ψ(t, x) satisfies the one-dimensional Schrodinger equation, and one or more integration by parts. 10. How to calculate averages such as ⟨x⟩, ⟨x2 ⟩, ⟨p⟩ = m d⟨x⟩ , dt ⟨p2 ⟩ = ⟨ Ψ |(−iℏ∂x )2 Ψ ⟩, (1) given some explicit wave function Ψ(t, x), and so how to calculate the standard deviations σx = √ ⟨x2 ⟩ − ⟨x⟩2 and similarly for σp . 11. How to draw qualitatively correct wave functions of bound states with diﬀerent energy values for a a variety of one-dimensional potential V (x), and how to draw qualitative potentials consistent with some given wave function or some given probability density. 12. How to work with a Dirac delta function, both in an integral and when solving the Schrodinger equation for a potential V that involves a delta function. 13. How to find the time-dependent solution Ψ(t, x) of the Schrodinger equation corresponding to some initial state Ψ(t0 , x) by expanding the initial state Ψ(t0 , x) = Ψ0 (x) in terms of the energy states ψn : Ψ0 (x) = ∞ ∑ cn ψn (x) where cn = ⟨ ψn | Ψ0 ⟩, (2) n=1 which then leads to the time-dependent state Ψ(t, x) = ∞ ∑ cn e−i(En /ℏ)t ψn (x). (3) n=1 You need to know how to interpret the physical meaning of the coeﬃcients cn = ⟨ ψn | Ψ0 ⟩, as “the amplitude to transition from the state on the right to the state on the left” when an energy measurement is carried out, and that |cn |2 is the probability of an energy measurement of a system in the state Ψ yielding the value En . ∑∞ You should know how to prove that n=1 |cn |2 = 1 in Eq. (3). You should know how to show that the average energy of the state Ψ(t, x) is given by ⟨E⟩ = ∞ ∑ En |cn |2 . (4) n=1 14. You should understand the implications of “collapse of the wave function” when a measurement is carried out: not only does an experiment of a certain observable (position, energy, momentum, etc) yield a particular value of the observable, the measurement also changes (collapses) the state of the system to a wave function corresponding to the particular observable. So a position measurement that yields the value x0 collapses the wave function Ψ(t, x) to the state δ(x − x0 ), an energy measurement with value En collapses the wave function to ψn (x), a momentum measurement that yields the value p0 collapses the wave function to ei(p/ℏ)x , and so on. The collapsed wave function then has a profound eﬀect on what happens when a new measurement of the same system is carried out. 15. You should understand how the fermionic nature of electrons in a many-electron system leads to the stacking of electrons in successive energy levels, with at most two electrons per energy level. From this, you should be able to predict what are the wavelengths (or frequencies) of absorption peaks of a many-electron system with given energy levels. For a conjugated organic molecule with N carbon atoms, you should be able to predict roughly where absorption peaks will occur. 2 Practice problems 1. French and Taylor: (a) 2-6, 2-8 . (b) 3-1, 3-3, 3-8, 3-11, 3-15 parts (b), (d), and (e), 3-16 parts (b), (c), and (d), and especially 3-18 on page 153. 2. If the brackets ⟨f (x)⟩ denotes the average of some function over an interval [a, b]: ⟨f (x)⟩ = show that 3. For the wave function 1 b−a ∫ b f (x) dx, (5) ⟨ ⟩ ⟨ ⟩ (f − ⟨f ⟩)2 = f 2 − ⟨f ⟩2 . (6) Ψ(t, x) = Ae−c|x| e−iωt , (7) a normalize Ψ and calculate the averages (expectation values) ⟨x⟩, ⟨p⟩, and ⟨x2 ⟩. 4. A particle of mass m is in a state described by the wave function Ψ(t, x) = Ae−a[(mx 2 /ℏ)+it] , (8) where A and a are real constants. Find the constant A so that Ψ is normalized, and determine what is the potential V (x) that appears in the Schrodinger equation. 5. For the one-dimensional Schrodinger equation with potential V (x) with bound state Ψ(t, x), show that the average energy ⟨E⟩ at time t is given by ⟨E⟩ = ⟨ Ψ | (iℏ∂t )Ψ ⟩ (9) given that Ψ can be written in terms of all the bound-state stationary solutions ψn (x) as Ψ(t, x) = ∞ ∑ cn e−i(En /ℏ)t ψn (x). (10) n=1 6. What is the value of ∫∞ 0 [cos(3x) + 2] δ(x − π) dx? 7. What are the wave functions ψn (x) and energies En for a particle of mass m that moves freely (V = 0) in a one-dimensional circular tube of radius R? This corresponds roughly to a bead of mass m that can move without friction on a ring of wire. 8. Consider a potential V (x) consisting of three delta functions: [ ] V (x) = −α δ(x + L) + δ(x − L) + βδ(x), (11) consisting of two negative spikes at x = ±L and one positive spike at x = 0. Here α > 0 and β > 0. (a) Draw the qualitative form of the wavefunction ψ(x) for the lowest energy bound state E1 and the second lowest energy bound state E2 (assuming the latter exists). (b) Give the mathematical description of the wave function ψ1 (x) for all regions of space. 3 (c) Derive and give necessary and suﬃcient conditions to determine the wave function over all of space. (d) Derive and give a transcendental equation whose solutions determine the lowest energy E1 . 9. A particle in an infinite square well of width L has the initial state Ψ(t = 0, x) = A sin3 (πx/L) . (12) (a) Calculate ⟨x⟩ as a function of time. (b) At time t, what is the probability of an energy measurement giving the value E2 ? (c) What is the average energy of this system at time t? 10. Consider the potential V (x) given by { V (x) = αδ(x) for −L < x < L, , ∞ for |x| > L (13) consisting of an infinitely tall well of width L with a delta-function peak at its center (so α > 0) (a) Sketch the wave functions ψ1 (x) and ψ2 (x) of the two lowest energy states and describe how they diﬀer from the corresponding wavefunctions when the delta function is absent (α = 0). (b) Solve explicitly the time-independent Schrodinger equation for the stationary states ψn (x) and their corresponding energies En , and discuss how the wave functions and energies are aﬀected by the delta function. 11. For the wave dispersion relation ω = c1 k 2 + c2 k 4 , (a) What is the phase speed vϕ and group speed vg , and which is larger in the limit of small wavelengths and in the limit of long wavelengths? (b) Derive a one-dimensional partial diﬀerential equation that wave functions u(t, x) consistent with this dispersion relation must satisfy. 12. Find and give expressions for the approximate wavelengths λ1 and λ2 of the two lowest energy absorption peaks of a conjugated carbon molecule consisting of three carbon atoms. The following pages contains some relevant questions taken from older physics GRE exams. 4