1. For the distribution of ages in Section 1.3.1: (a) Compute ...
(b) Determine ... for each j, and use Equation 1.11 to compute the
standard deviation. (c) Use your results in (a) and (b) to check
Equation 1.12. ... ... Get solution
2. (a) Find the standard deviation of the distribution in Example 1.1. (b) What is the probability that a photograph, selected at random, would show a distance x more than one standard deviation away from the average? Example 1.1: ... Get solution
3. Consider the gaussian distribution ... where A, a, and ...are positive real constants. (Look up any integrals you need.) (a) Use Equation 1.16 to determine A. (b) Find ... (c) Sketch the graph of ... ... Get solution
4. At time t = 0 a particle is represented by the wave function...where A, a, andb are constants.(a) Normalize Ψ (that is, find A, in terms of a and b).(b) Sketch Ψ(x, 0), as a function of x.(c) Where is the particle most likely to be found, at t = 0?(d) What is the probability of finding the particle to the left of a? Check your result in the limiting cases b = a and b = 2a.(e) What is the expectation value of x? Get solution
5. Consider the wave function...where A, λ, and ω are positive real constants. (We'll see in Chapter 2 what potential (V) actually produces such a wave function.)(a) Normalize Ψ.(b) Determine the expectation values of x and x2.(c) Find the standard deviation of x. Sketch the graph of |Ψ|2, as a function of x, and mark the points (...x... + σ) and (...x... — σ), to illustrate the sense in which σ represents the “spread” in x. What is the probability that the particle would be found outside this range? Get solution
6. Why can't you do integration-by-parts directly on the middle expression in Equation 1—pull the time derivative over onto x, note that ∂x/∂t = 0, and conclude that d...x.../dt = 0?Equation 1... Get solution
7. Calculate ... Answer: ... Equations 1.32 (or the first part of 1.33) and 1.38 are instances of Ehrenfest’s theorem. which tells us that expectation values obey classical laws. ... ... Get solution
8. Suppose you add a constant V0 to the potential energy (by “constant” I mean independent of x as well as t). In classical mechanics this doesn’t change anything, but what about quantum mechanics? Show that the wave function picks up a time-dependent phase factor: exp(-i V0t/ћ). What effect does this have on the expectation value of a dynamical variable? Get solution
9. A particle of mass m is in the state...where A and a are positive real constants.(a) Find A.(b) For what potential energy function V(x) does Ψ satisfy the Schrödinger equation?(c) Calculate the expectation values of x, x2, p, and p2.(d) Find σx and σp. Is their product consistent with the uncertainty principle? Get solution
10. Consider the first 25 digits in the decimal expansion of π (3, 1, 4, 1,5, 9,…).(a) If you selected one number at random, from this set, what are the probabilities of getting each of the 10 digits?(b) What is the most probable digit? What is the median digit? What is the average value?(c) Find the standard deviation for this distribution. Get solution
11. The needle on a broken car speedometer is free to swing, and bounces perfectly off the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and π.(a) What is the probability density, ρ(θ)? Hint: ρ(θ) dθ is the probability that needle will come to rest between θ and (θ + dθ). Graph ρ(θ) as a function of θ, from −π/2 to 3π/2. (Of course, part of this interval is excluded, so ρ is zero there.) Make sure that the total Probability is 1.(b) Compute ...θ..., ...θ2..., and σ, for this distribution.(c) Compute ...sinθ..., ...cosθ..., and ...cos2θ.... Get solution
12. We consider the same device as the previous problem, but this time we are interested in the x-coordinate of the needle point—that is, the “shadow,” or “projection,” of the needle on the horizontal line.(a) What is the probability density ρ(x)? Graph ρ(x) as a function of x, from −2r to + 2r, where r is the length of the needle. Make sure the total probability is 1. Hint: ρ(x) dx is the probability that the projection lies between x and (x + dx). You know (from Problem 1) the probability that θ is in a given range; the question is, what interval dx corresponds to the interval dθ?(b) Compute ...x..., ...x2..., and /, for this distribution. Explain how you could have obtained these results from part (c) of Problem 1.Problem 1The needle on a broken car speedometer is free to swing, and bounces perfectly off the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and π.(a) What is the probability density, ρ(θ)? Hint: ρ(θ) dθ is the probability that needle will come to rest between θ and (θ + dθ). Graph ρ(θ) as a function of θ, from −π/2 to 3π/2. (Of course, part of this interval is excluded, so ρ is zero there.) Make sure that the total Probability is 1.(b) Compute ...θ..., ...θ2..., and σ, for this distribution.(c) Compute ...sinθ..., ...cosθ..., and ...cos2θ.... Get solution
13. Buffon’s needle. A needle of length l is dropped at random onto a sheet of paper ruled with parallel lines a distance l apart. What is the probability that the needle will cross a line? Hint: Refer to Problem 1.Problem 1We consider the same device as the previous problem, but this time we are interested in the x-coordinate of the needle point—that is, the “shadow,” or “projection,” of the needle on the horizontal line.(a) What is the probability density ρ(x)? Graph ρ(x) as a function of x, from −2r to + 2r, where r is the length of the needle. Make sure the total probability is 1. Hint: ρ(x) dx is the probability that the projection lies between x and (x + dx). You know (from Problem 1) the probability that θ is in a given range; the question is, what interval dx corresponds to the interval dθ?(b) Compute ...x..., ...x2..., and σ, for this distribution. Explain how you could have obtained these results from part (c) of Problem 1.Problem 1The needle on a broken car speedometer is free to swing, and bounces perfectly off the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and π.(a) What is the probability density, ρ(θ)? Hint: ρ(θ) dθ is the probability that needle will come to rest between θ and (θ + dθ). Graph ρ(θ) as a function of θ, from −π/2 to 3π/2. (Of course, part of this interval is excluded, so ρ is zero there.) Make sure that the total Probability is 1.(b) Compute ...θ..., ...θ2..., and σ, for this distribution.(c) Compute ...sinθ..., ...cosθ..., and ...cos2θ.... Get solution
14. Let Pab(t) be the probability of finding a particle in the range (a at time t.(a) Show that...where...What are the units of J(x, t)? Comment: J is called the probability current, because it tells you the rate at which probability is “flowing” past the point x. If Pab(t) is increasing, then more probability is flowing into the region at one end than flows out at the other.(b) Find the probability current for the wave function in Problem 1. (This is not a very pithy example, I’m afraid; we’ll encounter more substantial ones in due course.)Problem 1A particle of mass m is in the state...where A and a are positive real constants.(a) Find A.(b) For what potential energy function V(x) does Ψ satisfy the Schrödinger equation?(c) Calculate the expectation values of x, x2, p, and p2.(d) Find σx and σp. Is their product consistent with the uncertainty principle? Get solution
15. Suppose you wanted to describe an unstable particle, that spontaneously disintegrates with a “lifetime” τ. In that case the total probability of finding the particle somewhere should not be constant, but should decrease at (say) an exponential rate:...A crude way of achieving this result is as follows. In Equation 1 we tacitly assumed that V (the potential energy) is real. That is certainly reasonable, but it leads to the “conservation of probability” enshrined in Equation 2. What if we assign toV an imaginary part:...where V0 is the true potential energy and Γ is a positive real constant?(a) Show that (in place of Equation 2) we now get...(b) Solve for P(t), and find the lifetime of the particle in terms of Γ.Equation 1...Equation 2... Get solution
16. Show that...for any two (normalizable) solutions to the Schrödinger equation, Ψ1, and Ψ2. Get solution
17. A particle is represented (at time t = 0) by the wave function...(a) Determine the normalization constant A.(b) What is the expectation value of x (at time t = 0)?(c) What is the expectation value ofp (at time t = 0)? (Note that you cannot get it from p = md...x.../dt. Why not?)(d) Find the expectation value of x2.(e) Find the expectation value of p2.(f) Find the uncertainty in x (σx).(g) Find the uncertainty in p (σp).(h) Check that your results are consistent with the uncertainty principle. Get solution
18. In general, quantum mechanics is relevant when the de Broglie wavelength of the particle in question (h/p) is greater than the characteristic size of the system (d). In thermal equilibrium at (Kelvin) temperature T, the average kinetic energy of a particle is...(where kB is Boltzmann’s constant), so the typical de Broglie wavelength is... [1.41]The purpose of this problem is to anticipate which systems will have to be treated quantum mechanically, and which can safely be described classically.(a) Solids. The lattice spacing in a typical solid is around d = 0.3 nm. Find the temperature below which the free18 electrons in a solid are quantum mechanical. Below what temperature are the nuclei in a solid quantum mechanical? (Use sodium as a typical case.) Moral: The free electrons in a solid are always quantum mechanical; the nuclei are almost never quantum mechanical. The same goes for liquids (for which the interatomic spacing is roughly the same), with the exception of helium below 4 K.(b) Gases. For what temperatures are the atoms in an ideal gas at pressure P quantum mechanical? Hint: Use the ideal gas law (PV = NkBT) to deduce the interatomic spacing. Answer: TkB)(h2/3m)3/5 P2/5 Obviously (for the gas to show quantum behavior) we wantm to be as small as possible, and P as large as possible. Put in the numbers for helium at atmospheric pressure. Is hydrogen in outer space (where the interatomic spacing is about 1 cm and the temperature is 3 K) quantum mechanical? Get solution
2. (a) Find the standard deviation of the distribution in Example 1.1. (b) What is the probability that a photograph, selected at random, would show a distance x more than one standard deviation away from the average? Example 1.1: ... Get solution
3. Consider the gaussian distribution ... where A, a, and ...are positive real constants. (Look up any integrals you need.) (a) Use Equation 1.16 to determine A. (b) Find ... (c) Sketch the graph of ... ... Get solution
4. At time t = 0 a particle is represented by the wave function...where A, a, andb are constants.(a) Normalize Ψ (that is, find A, in terms of a and b).(b) Sketch Ψ(x, 0), as a function of x.(c) Where is the particle most likely to be found, at t = 0?(d) What is the probability of finding the particle to the left of a? Check your result in the limiting cases b = a and b = 2a.(e) What is the expectation value of x? Get solution
5. Consider the wave function...where A, λ, and ω are positive real constants. (We'll see in Chapter 2 what potential (V) actually produces such a wave function.)(a) Normalize Ψ.(b) Determine the expectation values of x and x2.(c) Find the standard deviation of x. Sketch the graph of |Ψ|2, as a function of x, and mark the points (...x... + σ) and (...x... — σ), to illustrate the sense in which σ represents the “spread” in x. What is the probability that the particle would be found outside this range? Get solution
6. Why can't you do integration-by-parts directly on the middle expression in Equation 1—pull the time derivative over onto x, note that ∂x/∂t = 0, and conclude that d...x.../dt = 0?Equation 1... Get solution
7. Calculate ... Answer: ... Equations 1.32 (or the first part of 1.33) and 1.38 are instances of Ehrenfest’s theorem. which tells us that expectation values obey classical laws. ... ... Get solution
8. Suppose you add a constant V0 to the potential energy (by “constant” I mean independent of x as well as t). In classical mechanics this doesn’t change anything, but what about quantum mechanics? Show that the wave function picks up a time-dependent phase factor: exp(-i V0t/ћ). What effect does this have on the expectation value of a dynamical variable? Get solution
9. A particle of mass m is in the state...where A and a are positive real constants.(a) Find A.(b) For what potential energy function V(x) does Ψ satisfy the Schrödinger equation?(c) Calculate the expectation values of x, x2, p, and p2.(d) Find σx and σp. Is their product consistent with the uncertainty principle? Get solution
10. Consider the first 25 digits in the decimal expansion of π (3, 1, 4, 1,5, 9,…).(a) If you selected one number at random, from this set, what are the probabilities of getting each of the 10 digits?(b) What is the most probable digit? What is the median digit? What is the average value?(c) Find the standard deviation for this distribution. Get solution
11. The needle on a broken car speedometer is free to swing, and bounces perfectly off the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and π.(a) What is the probability density, ρ(θ)? Hint: ρ(θ) dθ is the probability that needle will come to rest between θ and (θ + dθ). Graph ρ(θ) as a function of θ, from −π/2 to 3π/2. (Of course, part of this interval is excluded, so ρ is zero there.) Make sure that the total Probability is 1.(b) Compute ...θ..., ...θ2..., and σ, for this distribution.(c) Compute ...sinθ..., ...cosθ..., and ...cos2θ.... Get solution
12. We consider the same device as the previous problem, but this time we are interested in the x-coordinate of the needle point—that is, the “shadow,” or “projection,” of the needle on the horizontal line.(a) What is the probability density ρ(x)? Graph ρ(x) as a function of x, from −2r to + 2r, where r is the length of the needle. Make sure the total probability is 1. Hint: ρ(x) dx is the probability that the projection lies between x and (x + dx). You know (from Problem 1) the probability that θ is in a given range; the question is, what interval dx corresponds to the interval dθ?(b) Compute ...x..., ...x2..., and /, for this distribution. Explain how you could have obtained these results from part (c) of Problem 1.Problem 1The needle on a broken car speedometer is free to swing, and bounces perfectly off the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and π.(a) What is the probability density, ρ(θ)? Hint: ρ(θ) dθ is the probability that needle will come to rest between θ and (θ + dθ). Graph ρ(θ) as a function of θ, from −π/2 to 3π/2. (Of course, part of this interval is excluded, so ρ is zero there.) Make sure that the total Probability is 1.(b) Compute ...θ..., ...θ2..., and σ, for this distribution.(c) Compute ...sinθ..., ...cosθ..., and ...cos2θ.... Get solution
13. Buffon’s needle. A needle of length l is dropped at random onto a sheet of paper ruled with parallel lines a distance l apart. What is the probability that the needle will cross a line? Hint: Refer to Problem 1.Problem 1We consider the same device as the previous problem, but this time we are interested in the x-coordinate of the needle point—that is, the “shadow,” or “projection,” of the needle on the horizontal line.(a) What is the probability density ρ(x)? Graph ρ(x) as a function of x, from −2r to + 2r, where r is the length of the needle. Make sure the total probability is 1. Hint: ρ(x) dx is the probability that the projection lies between x and (x + dx). You know (from Problem 1) the probability that θ is in a given range; the question is, what interval dx corresponds to the interval dθ?(b) Compute ...x..., ...x2..., and σ, for this distribution. Explain how you could have obtained these results from part (c) of Problem 1.Problem 1The needle on a broken car speedometer is free to swing, and bounces perfectly off the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and π.(a) What is the probability density, ρ(θ)? Hint: ρ(θ) dθ is the probability that needle will come to rest between θ and (θ + dθ). Graph ρ(θ) as a function of θ, from −π/2 to 3π/2. (Of course, part of this interval is excluded, so ρ is zero there.) Make sure that the total Probability is 1.(b) Compute ...θ..., ...θ2..., and σ, for this distribution.(c) Compute ...sinθ..., ...cosθ..., and ...cos2θ.... Get solution
14. Let Pab(t) be the probability of finding a particle in the range (a at time t.(a) Show that...where...What are the units of J(x, t)? Comment: J is called the probability current, because it tells you the rate at which probability is “flowing” past the point x. If Pab(t) is increasing, then more probability is flowing into the region at one end than flows out at the other.(b) Find the probability current for the wave function in Problem 1. (This is not a very pithy example, I’m afraid; we’ll encounter more substantial ones in due course.)Problem 1A particle of mass m is in the state...where A and a are positive real constants.(a) Find A.(b) For what potential energy function V(x) does Ψ satisfy the Schrödinger equation?(c) Calculate the expectation values of x, x2, p, and p2.(d) Find σx and σp. Is their product consistent with the uncertainty principle? Get solution
15. Suppose you wanted to describe an unstable particle, that spontaneously disintegrates with a “lifetime” τ. In that case the total probability of finding the particle somewhere should not be constant, but should decrease at (say) an exponential rate:...A crude way of achieving this result is as follows. In Equation 1 we tacitly assumed that V (the potential energy) is real. That is certainly reasonable, but it leads to the “conservation of probability” enshrined in Equation 2. What if we assign toV an imaginary part:...where V0 is the true potential energy and Γ is a positive real constant?(a) Show that (in place of Equation 2) we now get...(b) Solve for P(t), and find the lifetime of the particle in terms of Γ.Equation 1...Equation 2... Get solution
16. Show that...for any two (normalizable) solutions to the Schrödinger equation, Ψ1, and Ψ2. Get solution
17. A particle is represented (at time t = 0) by the wave function...(a) Determine the normalization constant A.(b) What is the expectation value of x (at time t = 0)?(c) What is the expectation value ofp (at time t = 0)? (Note that you cannot get it from p = md...x.../dt. Why not?)(d) Find the expectation value of x2.(e) Find the expectation value of p2.(f) Find the uncertainty in x (σx).(g) Find the uncertainty in p (σp).(h) Check that your results are consistent with the uncertainty principle. Get solution
18. In general, quantum mechanics is relevant when the de Broglie wavelength of the particle in question (h/p) is greater than the characteristic size of the system (d). In thermal equilibrium at (Kelvin) temperature T, the average kinetic energy of a particle is...(where kB is Boltzmann’s constant), so the typical de Broglie wavelength is... [1.41]The purpose of this problem is to anticipate which systems will have to be treated quantum mechanically, and which can safely be described classically.(a) Solids. The lattice spacing in a typical solid is around d = 0.3 nm. Find the temperature below which the free18 electrons in a solid are quantum mechanical. Below what temperature are the nuclei in a solid quantum mechanical? (Use sodium as a typical case.) Moral: The free electrons in a solid are always quantum mechanical; the nuclei are almost never quantum mechanical. The same goes for liquids (for which the interatomic spacing is roughly the same), with the exception of helium below 4 K.(b) Gases. For what temperatures are the atoms in an ideal gas at pressure P quantum mechanical? Hint: Use the ideal gas law (PV = NkBT) to deduce the interatomic spacing. Answer: TkB)(h2/3m)3/5 P2/5 Obviously (for the gas to show quantum behavior) we wantm to be as small as possible, and P as large as possible. Put in the numbers for helium at atmospheric pressure. Is hydrogen in outer space (where the interatomic spacing is about 1 cm and the temperature is 3 K) quantum mechanical? Get solution
