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Assignments for Chapter 17 The Particle in the Box and in the Real World Deadline: Before the end of the next chapter. 下一章結束日或之前繳交。 Questions on Concepts You hand in the answers to any 10 or more of the following questions: 劃了藍線底線的題目不要求。 Q17.1 Why does the Stern‐Gerlach experiment show that the operator “ measure the z component of the magnetic moment of an Ag atom” has only two eigenfunctions with eigenvalues that have the same magnitude and opposite sign? Q17.2 Have a closer look at Equation (17.6) and Figure 17.4. How would Figure 17.4 change if m increases? Generalize your conclusion to make a statement of how well the momentum is known if the position is known exactly. Q17.3 Why is maintaining the entanglement of pairs A and B and A and X the crucial ingredient of teleportation? Q17.4 Why is it not possible to reconstruct the wave function of a quantum mechanical superposition state from experiments? Q17.5 Why does the relative uncertainty in x for the particle in the box increase as n → ∞ ? Q17.6 Why is the statistical concept of variance a good measure of uncertainty in a quantum mechanical measurement? Q17,7 [ ]
[ ]
Derive a relationship between Aˆ , Bˆ and Bˆ , Aˆ . Q17.8 How does our study of the eigenfunctions for the particle in the box let us conclude that the position uncertainty has its minimum value for n=1? Q17.9 What is the difference between a bit and a qubit? Q17.10 How does the Heisenberg uncertainty principle allow us to conclude that it is not possible to make exact copies of quantum mechanical objects? Q17.11 Which result of the Stern‐Gerlach experiment allows us to conclude that the operator for the z and x components of the magnetic moment do not commute? Q17.12 Why is it impossible to determine the state of a quantum mechanical system exactly by measurement? Q17.13 Why is the motion of a human being not described by the Schrodinger equation rather than Newton’s second law if every atom in our body is described by quantum mechanics? Q17.14 Explain the following statement: if = = 0 , it would be possible to measure the position and momentum of a particle exactly and simultaneously. Q17.15 Why is p 2 rather than p used to calculate the relative uncertainty for the particle in the box? _____________________________________________________________________ Problems You hand in the answers to any 10 or more of the following questions: 劃了藍線底線的題目不要求。 P17.1 In this problem, we consider the calculations for σ p and σ x for the particle in the box shown in Figure 17.5 in more detail. In particular we want to determine how the absolute uncertainty Δp x and the relative uncertainty Δp x / p x of a single peak corresponding to either the most probable positive or negative momentum depends on the quantum number n. a. First we must relate k and p x . From E = p x2 / 2m and E = n 2 h 2 / 8ma 2 , show that p x = nh / 2a . b. Use the result from part (a) together with the relation linking the length of the box and the allowed wavelengths to obtain p x = =k c. Relate Δp x and Δp x / p x with k and Δk . d. The following graph shows Ak versus k − k peak . By plotting the results of 2
Figure 17.5 in this way, all peaks appear at the same value of the abscissa. Successive 2
curves have been shifted upward to avoid overlap. Use the width of the Ak peak at half height as a measure of Δk . What can you conclude from this graph about the dependence of Δp x on n? 2
e. The following graph shows Ak versus k/n for n=3,n=5,n=7,n=11,n=31,and n=101. Use the width of the Ak peak at half height as a measure of Δk / n . Using 2
the graphs, determine the dependence of Δp x / p x on n. One way to do this is to assume that the width depends on n like (Δp x / p x ) = n α where α is a constant to be determined. If this relationship holds, a plot of ln(Δp x / p x ) versus ln n will be linear and the slope will give the constant α . P17.2 Consider the results of Figure 17.5 more quantitatively. Describe the values of x and k by x ± Δx and k 0 ± Δk . Evaluate Δx from the zero of distance to the point at which the envelope of Ψ ∗ ( x)Ψ ( x) is reduced to one‐half of its peak value. Evaluate Δk from Δk = 1 / 2(k 0 − k min ) where k 0 is the average wavevector of the set of 21 waves (11th of 21) and k min corresponds to the 21st of the 21 waves. Is your estimated value of ΔpΔx = =ΔkΔx in reasonable agreement with the Heisenberg uncertainty principle? P17.3 Evaluate the commutator [d / dr,1 / r ] by applying the operators to an arbitrary function f (r ) . P17.4 Show a. that Ψ( x) = e − x
2
/2
is an eigenfunction of Aˆ = x 2 − ∂ 2 / ∂x 2 ; and b. that Bˆ Ψ ( x ) (where Bˆ = x − ∂ / ∂x ) is another eigenfunction of  . P17.5 Another important uncertainty principle is encountered in time‐dependent systems. It relates the lifetime of a state Δt with the measure spread in the photon energy ΔE associated with the decay of this state to a stationary state of the system. “Derive” the relation ΔEΔt ≥ = / 2 in the following steps. a. Starting from E = p x2 / 2m and ΔE = (dE / dp x )Δp x , show that ΔE = v x Δp x . b. Using v x = Δx / Δt , show that ΔEΔt = Δp x Δx ≥ = / 2 . c. Estimate the width of a spectral line originating from the decay of a state of lifetime 1.0 × 10 −9 s and 1.0 × 10 −11 s in inverse seconds and inverse centimeters. P17.6 Evaluate the commutator [xˆ (∂ / ∂y ), yˆ ] by applying the operators to an arbitrary function f ( x, y ) . P17.7 [ ]
Evaluate Aˆ , Bˆ if Aˆ = x 2 − d 2 / dx 2 and Bˆ = x − d / dx . P17.8 Consider the entangled wave function for two photons, 1
(Ψ1 ( H )Ψ2 (V ) + Ψ1 (V )Ψ2 ( H )) Ψ12 =
2
Assume that the polarization operator p̂ i has the properties Pˆi Ψi (H ) = −Ψi ( H ) and Pˆi Ψi (V ) = +Ψi (V ) where i = 1 or i = 2 . a. Show that Ψ12 is not an eigenfunction of p̂1 and p̂ 2 . b. Show that each of the two terms in Ψ12 is an eigenfunction of the polarization operator p̂1 . c. What is the average value of the polarization P1 that you will measure on identically prepared systems? P17.9 [
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Evaluate the commutator pˆ x , pˆ x2 by applying the operator to an arbitrary function f (x ) . P17.10 Revisit the double‐slit experiment of Example Problem 17.2. Using the same geometry and relative uncertainty in the momentum, what electron momentum would give a position uncertainty of 1.0 × 10 −9 m? What is the ratio of the wavelength and the slit spacing for this momentum? Would you expect a pronounced diffraction effect for this wavelength? P17.11 Evaluate the commutator d 2 / dx 2 , x by applying the operators to an arbitrary function f (x ) . [
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P17.12 Revisit the TV picture tube of Example Problem 17.3.Keeping all other parameters the same, what electron energy would result in a position uncertainty of 1.00 × 10 −8 m along the direction of motion? P17.13 [
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Evaluate the commutator (d 2 / dx 2 ) − x, (d / dx) + x 2 by applying the operators to an arbitrary function f (x ) . P17.14 If the wave function describing a system is not an eigenfunction of the operator B̂ , measurements on identically prepared systems will give different result. The variance of this set of results is defined in error analysis as σ B2 = (B − B
)
2
, where B is the value of the observable in a single measurement and B is the average of all measurements. Using the definition of average value from the quantum mechanical postulates, A = ∫ Ψ ∗ ( x) Aˆ Ψ ( x)dx , show that σ B2 = B 2 − B . 2
P17.15 Apply the Heisenberg uncertainty principle to estimate the zero point energy for the particle in the box. a. First justify the assumption that Δx ≤ a and that, as a result, Δp ≥ = / 2a . Justify the statement that, if Δp ≥ 0 , we cannot know that E = p 2 / 2m is identically zero. b. Make this application more quantitative. Assume that Δx = 0.50a and Δp = 0.5 p where p is the momentum in the lowest energy state. Calculate the total energy of this state based on these assumptions and compare your result with the ground‐state energy for the particle in the box. c. Compare your estimates for Δp and Δx with the more rigorously derived uncertainties σ p and σ x of Equation (17.13). P17.16 Evaluate the commutator d / dx, x 2 by applying the operators to an arbitrary function f (x ) . [
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P17.17 Evaluate the commutator [xˆ , pˆ x ] by applying the operators to an arbitrary function f (x ) . What value does the commutator [ pˆ x , xˆ ] have? P17.18 In this problem, you will carry out the calculations that describe the Stern‐Gerlach experiment shown in Figure 17.2. Classically, a magnetic dipole μ has the potential
energy E=-μ․B. If the field has a gradient in the z direction, the magnetic moment will
experience a force, leading it to be deflected in the z direction. Because classically μ can
take on any value in the range − μ ≤ μ z ≤ μ , a continuous range of positive and
negative z deflections of a beam along the y direction will be observed. From a quantum
mechanical perspective, the forces are the same as in the classical picture, but μ z can
only take on a discrete set of values. Therefore, the incident beam will be split into a
discrete set of beams that have different deflections in the z direction.
a. The geometry of the experiment is shown here. In the region of the magnet indicated by
d1 , the Ag atom experiences a constant force. It continues its motion in the force-free
region indicated by d 2 . If the force inside the magnet is Fz , show that
z = 1 / 2(Fz / m Ag )t12 + t 2 v z (t1 ) . The flight times t1 and t 2 correspond to the regions
d1 and d 2 .
1 2
⎛
⎜ d1 d 2 + d1
2
b. Show that assuming a small deflection, z = Fz ⎜
⎜ m Ag v y2
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎠
c. The magnetic moment of the electron is given by μ = gsμ B / 2. In this equation, μ B
is the Bohr magneton and has the value 9.274 × 10 −24 J / tesla . The gyromagnetic ratio of
the electron gs has the value 2.00231. If ∂Bz / ∂z = 1000 tesla m‐1, and d1 and d 2
are 0.200 and 0.250 m, respectively, and v y = 500.ms −1 , what values of z will be observed? P17.19 Evaluate the commutator [(d / dx ) − x, (d / dx) + x] by applying the operators to an arbitrary function f ( x ) P17.20 [
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Evaluate the commutator xˆ , pˆ x2 by applying the operators to an arbitrary function f ( x) P17.21 What is wrong with the following argument? We know that the functions Ψn ( x) = 2 / a sin(nπx / a) are eigenfunctions of the total energy operator for the particle in the infinitely deep box. We also know that in the box, E = p x2 / 2m + V ( x) = p x2 / 2m . Therefore, the operator for Etotal is proportional to 2
2
the operator for p x . Because the operator for p x and p x commute as you demonstrated in Problem P17.8, the functions Ψn ( x) = 2 / a sin(nπx / a) are eigenfunctions of both the total energy and momentum operators. P17.22 [
] [ ]
[ ]
ˆ ˆˆ
ˆ ˆ ˆ ˆ ˆ ˆ
For linear operators A, B, and C, show that A, BC = A, B C + B A, C . P17.23 The muzzle velocity of a rifle bullet is about 900. ms‐1 along the direction of motion. If the bullet weighs 30. g, and the uncertainty in its momentum is 0.10%, how accurately can the position of the bullet be measured along the direction of motion? 下述題目不要求。 Web‐Based Simulations, Animations, and Problems W17.1 The simulation of particle diffraction from a single slit is used to illustrate the dependence between the uncertainty in the position and momentum. The slit width and particle velocity are varied using sliders. W17.2 The Heisenberg uncertainty principle states that ΔpΔx > = / 2 . In an experiment, it is more likely that λ is varied rather than p, where λ is the de Broglie wavelength of the particle. The relationship between Δx and Δλ will be determined using a simulation. Δx will be measured as a function of Δλ at a constant value of λ, and as a function of λ for a constant value of Δλ . W17.3 The uncertainty in momentum will be determined for the total energy eigenfunctions for the particle in the infinite depth box for several values of the quantum number n. The function describing in k, g n (k ) =
1
2π
∞
∫
f n ( x)e −ikx dx =
−∞
1
2π
a
∫ sin
0
nπx −ikx
e dx a
will be determined. The values of k for which this function has maxima will be compared with that expected for a classical particle of momentum p = 2mE . The width in k of the function g n (k ) on n will be investigated.