YHJ - XJ\L2 ] (c) Compute XSin[J]\ and

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1 Homework # Physics 4 - Spring 5 Due Friday, January 3, 5 Read: B&J Chapters - 3. Determine the energy in ev (electron volts) of (a) a Γ-ray photon with wavelength Λ = -5 m (b) an x-ray photon with wavelength Λ = Þ (c) a visible photon with wavelength Λ = 35 Þ. What color is this photon?. Using the uncertainty principle DxD px ³ Ñ, determine the uncertainty of the momentum and the average px + py + pz p kinetic energy [ _=[ _ of the following particles. Express the kinetic energy in ev s. Hint: By m m definition D px = YH px - X px \L ] = X px \ =, Y p y ] =, X pz \ =. Y px ] - X px \ and if a particle is confined to a sphere Dx~ Dy~ Dz~ r and (a) an electron at rest confined to a sphere of radius of the first Bohr orbit, a =.5 - m (b) a proton at rest confined to a sphere of radius of the first Bohr orbit, a =.5 - m (c) a proton at rest confined to a sphere of radius Fermi I -5 mm 3. Consider a dial indicator whose needle is free to rotate so that if you give the needle a flick it is equally likely to come to rest at any angle between and. (a) What is the probability density, Ρ@JD? Graph Ρ[J] as a function of J from to. [Note: Ρ@JD â J is the probability that the needle will come to rest between J and J + â J] (b) Compute XJ\, YJ ] and Σ = YHJ - XJ\L ] (c) Compute XSin[J]\ and 4. Using the same device as in Prob. 3, consider the x-coordinate of the needle point i.e. the projection of the needle on the horizontal axis. (a) What is the probability density, Ρ@xD. Graph Ρ@xD as a function of x from - r to r where r is the length of the needle. [Note: Ρ@xD â x is the probability that the projection lies between x and x + dx. Hint: From Prob. 3 you know the probability that J falls within a certain interval. What interval â x corresponds to the interval of â J. (b) Compute Xx\, Yx ] and Σ = YHx - Xx\L ] 5. A particle is defined in space by Ψ@x, td = : A ã-ä Hk x-ω tl where k and Ω are constants. I + x M (a) Find the normalization constant, A, up to an undetermined phase angle. (b) What is the probability of finding the particle in the interval x and x + dx at time t? Does this value depend on the value of t? (c) Sketch the probability density as a function of x. (d) Determine the uncertainty in position i.e. YHx - Xx\L ] 6. Using the wavefunction defined in Prob. 5 (a) Determine the k-space wavefunction Φ@kD and verify that it is normalized. To evaluate the integral, you may need to look it up, do a contour integral or use Mathematica. (b) Determine the average value of k using the momentum space wavefunction Φ@kD. (c) Determine the uncertainty in momentum i.e. YHk - Xk\L ]. (d) Does this wavefunction satisfies the uncertainty relationship D x D px > Ñ 7. Show by explicit integration that the following representation for the delta function does in fact agree with the -Hx-x L definition of the delta - x D = lim e. Hint: Multiply the delta function by a general function and expand the function in a Taylor series and integrate term by term. Then take the limit in the delta function definition.

2 7. Show by explicit integration that the following representation for the delta function does in fact agree with the -Hx-x L definition of the delta - x D = lim e. Hint: Multiply the delta function by a general function and expand the function in a Taylor series and integrate term by term. Then take the limit in the delta function definition. PHYS_4_HW Spring4.nb 8. By using the operational definition of the delta function dx = f HL where is a well-behaved - function, verify the following operational xd where a ¹ a '@- xd = - '@xd '@xd = - f where '@xd = - x = where the only zero of is at x i.e. D xd x Taylor expansion of f Hint: Consider the

3 PhotonEnergiesWavelengthColor_S5.nb Photon Energies, Wavelengths and Color Determine the energy in ev (electron volts) of (a) a Γ-ray photon with wavelength Λ. 5 m. (b) an x-ray photon with wavelength Λ. A. (c) a visible photon with wavelength Λ 35 A. What is the color of this photon. (a) a Γ-ray photon with wavelength Λ. 5 m. The energy of a photon is given by ε Ω ħ c k ħ Λ ħ c () where a very useful number to remember is ħc ev nm. Substituting ε Λ ħc ev nm. 5 m ev () (b) an x-ray photon with wavelength Λ. A. Substituting ε Λ ħc ev nm. A 699. ev (3) (c) a visible photon with wavelength Λ 35 A. What is the color of this photon. Substituting ε Λ and the color is ħc ev nm A ev (4)

4 PhotonEnergiesWavelengthColor_S5.nb A

5 Estimate Kinetic Energy from Uncertainty Relationship_S5.nb Estimating Kinetic Energy of a Particle Confined to a Sphere Using the uncertainty principle x p x ħ determine the uncertainty of the momentum and the average kinetic energy p m p x p y p z of the following particles. Express the kinetic m energy in ev's. Hint: By definition p x p x p x p x p x and if a particle is confined to a sphere of radius r then x y z r and p x p y p z.(a) an electron at rest confined to a sphere of radius, a 5. m. (b) a proton at rest confined to a sphere of radius, a 5. m. (c) a proton at rest confined to a sphere of radius r N Fermi. 5 m The average kinetic energy of the particle is just p m p x p y p z. From the definition of the m momentum uncertainty p x p x p x p x p x () we see that p x p x p x. Moreover, if the particle is confined to a sphere, its average momentum must be zero, p x so that p x p x. Therefore we can estimate the average kinetic energy from the uncertainty in the momenta. p p m x p y p z m p x p y p z m () If a particle is confined to a sphere of radius, r, then the uncertainty in the x direction is roughly x r. From the uncertainty relationship x p x ħ (3) we have that the uncertainty in the x momentum is p x ħ 4 r. Similarly for the uncertainty in the y and z directions, p y ħ 4 r and p z ħ 4 r. Therefore p m p x p y p z m m ħ 4 r ħ 4 r ħ 4 r 3 ħ 3 m r (4) One can rationalize the expression by multiplying both the top and bottom with c so that the average kinetic energy is p 3 ħc m 3 mc r (5) where ħc ev nm. (a) an electron at rest confined to a sphere of radius, a 5. m

6 Estimate Kinetic Energy from Uncertainty Relationship_S5.nb The uncertainty in the momentum, p x ħ 4 r can be written as c p x ħ c 4 r where ħc ev nm. For an electron localized to the first Bohr orbit, r.5 A.5 nm For an electron localized to a sphere of radius, a 5. m c p x ħc 4 r ev nm 4 5. m ev (6) or p x ħ 4 r ev s m kg m s (7) The average kinetic energy is then p 3 ħc m 3 mc r (8) where for an electron m c.5 MeV. Therefore p 3 ħ c m 3 m c r ev nm 3.5 MeV 5. m ev (9) As we shall see, this is comparable to the kinetic energy determine by solving Schrodinger equation for the hydrogen atom. (b) an proton at rest confined to a sphere of radius, a 5. m The uncertainty in the momentum, p x ħ for an proton localized to a sphere of radius, 4 r r n. 5 m has the same value as before c p x ħc 4 r ev nm 4 5. m ev () or p x ħ 4 r ev s m kg m s () However the mass of the proton is given by mc 938 MeV and so p 3 ħ c m 3 m c r ev nm MeV 5. m ev () As expected the larger mass causes the average kinetic energy to be greatly reduced. Given such a

7 Estimate Kinetic Energy from Uncertainty Relationship_S5.nb 3 large uncertainty in position compared to the size of a proton r p Fermi 5 m, the particle can be treated classically. (c) a proton at rest confined to a sphere of radius, r N. 5 m Now the uncertainty in the momentum, p x ħ 4 r becomes much larger c p x ħc 4 r ev nm 4. 5 m ev (3) or p x ħ 4 r.8765 ev s m.38 kg m s (4) Now the average kinetic energy is typical of a nuclear system. p 3 ħ c m 3 m c r ev nm MeV. 5 m.9793 MeV (5)

8 NeedleProbability_Theta_x_S5.nb Dial Indicator - Probability density Ρ[ϑ] Consider a dial indicator whose needle is free to rotate so that if you give the needle a flick it is equally likely to come to rest at any angle between and. (a) What is the probability density, Ρ ϑ? Graph Ρ[ϑ] as a function of ϑ from to. [Note: Ρ ϑ ϑ is the probability that the needle will come to rest between ϑ and ϑ ϑ] (b) Compute ϑ, ϑ and Σ (c) Compute Sin[ϑ] and Cos ϑ ϑ ϑ (a) What is the probability density, Ρ ϑ? Graph Ρ[ϑ] as a function of ϑ from to. If the needle is equally likely to come to rest at any angle between and, the probability density must be a constant. To determine the constant we use the fact that the probability density is normalize, i.e. the probability of finding it betwen and is unity. Ρ ϑ ϑ c ϑ c () and therefore c and the probability that the needle comes to rest in the interval between ϑ and ϑ is given by ϑ and ϑ Ρ ϑ ϑ ϑ ϑ () Plotting the probability density Ρ ϑ.5

9 NeedleProbability_Theta_x_S5.nb (b) Compute ϑ, ϑ and Σ ϑ ϑ The average value of any function of ϑ is determined by integrating that function times the probability Ρ x x over all possible values of ϑ ϑ. f f ϑ Ρ ϑ ϑ (3) Therefore the average of ϑ becomes ϑ ϑ ϑ Ρ ϑ ϑ ϑ ϑ (4) The average of ϑ becomes ϑ ϑ ϑ3 Ρ ϑ ϑ ϑ ϑ 3 3 (5) Using the relationship that ϑ ϑ ϑ ϑ we have that ϑ ϑ ϑ ϑ 3 3 (6) (c) Compute Sin[ϑ] and Cos ϑ Similarly, the average of Sin ϑ becomes Sin ϑ Sin ϑ Ρ ϑ ϑ Sin ϑ Cos ϑ ϑ (7) and the average of Cos ϑ is Cos ϑ Cos ϑ Ρ ϑ ϑ Cos ϑ ϑ Cos ϑ Sin ϑ ϑ (8)

10 NeedleProbability_x_Thet_5a.nb Dial Indicator - Probability density Ρ[x] Using the same device as in the previous problem, consider the x-coordinate of the needle point i.e. the projection of the needle on the horizontal axis. (a) What is the probability density Ρ@xD. Graph Ρ@xD as a function of x from - r to r where r is the length of the needle. [Note: ΡHxL â x is the probability that the projection lies between x and x + â x. Hint: From the previous you know the probability that J falls within a certain interval. What interval â x corresponds to the interval of â J. (b) Compute Xx\, Yx ] and Σ = YHx - Xx\L ] The probability that the needle comes to rest in the interval between J and â J is given by Ρ@JD â J = : J < and J > âj () J To find the probability that the projection of the needle falls between x and x + â x, we need to consider the relationship betwen x and J i.e. x = r Cos@JD. Because the probability that the needle comes to rest within the â J shown is â J, this must also be the probability that the needle falls within the corresponding â x. âj âx Therefore Ρ@xD â x = â J where â x = r Sin@JD â J () and Ρ@xD â x = âx r Sin@JD (3)

11 NeedleProbability_x_Thet_5a.nb Expressing in terms of x, = I - Cos@xD M = I - Hx rl M we obtain âx r I - Hx rl M Ρ@xD â x = (4) Checking the normalization -r x r à r -r - r x x ArcSinB F r âx = r = (5) -r r Plotting the funtion we note that the probability diverges at x = ±r...5 ΡHxL (b) Compute Xx\, Yx ] and Σ = YHx - Xx\L ] x r The average value of x is determined by integrating x times the probability Ρ HxL â x and integrating over all values of x H-r x rl. By symmetry this average should be zero. Xx\ = à r -r x âx r I - Hx rl M = The average value of x is determined by integrating x times the probability Ρ HxL â x and integrating over all values of x H-r x rl. For this case the average should not be zero (6)

12 NeedleProbability_x_Thet_5a.nb Yx ] = à x â x r -r r I - Hx rl M = r (7) YHx - Xx\L ] = and finally using the relationship that Σ = Σ= Yx ] - Xx\ = r - = r Yx ] - Xx\ we have that (8) 3

13 Particle Wavefunction in Coordinate Space A particle is defined in space by Ψ@x, td = : A ã-ä Hk x-ω tl where k and Ω are constants I + Hx al M (a) Find the normalization constant, A, up to an undetermined phase angle. (b) What is the probability of finding the particle in the interval x and x + dx at time t? Does this value depend on the value of t? (c) Sketch the probability density as a function of x. YHx - Xx\L ] (d) Determine the uncertainty in position i.e. (a) Find the normalization constant, A, up to an undetermined phase angle. To determine the normalization constant we require that à Ψ@x, td â x = () - Using the above form for Ψ@x, td we obtain 9 ΨHx, tl = = A ã-ä H-t Ω+x k L ãä H-t Ω+x k L A* + x + a x = A + a x () a Integrating à ΨHx, tl â x = à - A - + x âx (3) a This integral can be done by trig substition. Let x = a Tan@ΦD, â x = a Sec@ΦD â Φ with - Φ à ΨHx, tl â x = A à - - = A à - = A a a Sec@ΦD I + Tan@ΦD M Φ â Φ = A à a Cos@ΦD â Φ - Ha + a Cos@ ΦDL â Φ + 4 Sin@ ΦD - (4) = A a Setting this equal to, we can define A up to an undetermined phase angle j

14 Psi(x)_dx_5.nb A= a (5) äj ã For simplicity, we shall assume that A is real. (b) What is the probability of finding the particle in the interval x and x + â x at time t? Does this value depend on the value of t? The probability of finding the particle in the interval x and x + â x at time t is given by 9 ΨHx, tl = â x = A + x âx = a + a x âx (6) a which is clearly time independent. As you can see even though the wavefunction is time dependent, the probability density can be time independent. Note also that the probability is dimensionless as it should be. (c) Sketch the probability density as a function of x. Because the function includes an unknown parameter a, one must used scaled axes. Plotting the probability density is 9 ΨHx, tl = = A + x = a a + x (7) a.4.. a ΨHx, tl x a (d) Determine the uncertainty in position i.e. D x = YHx - Xx\L ] As was shown in class, the above expression can be reduced to 4

15 Psi(x)_dx_5.nb YHx - Xx\L ] = To calculate Xx\ we evaluate Yx - x Xx\ + Xx\ ] = à x 9 ΨHx, tl = â x = à - A Yx ] - Xx\ x - + x (8) âx = (9) a as expected from the symmetry of the probability distribution. à x 9 ΨHx, tl = â x = à - A x - + x âx = a à - a x + x âx () a This integral can be solved in various ways. One is to make the trig substitution x = a Tan@ΦD, â x = a Sec@ΦD â Φ with - Φ Yx ] = a à - = a à - = and so Dx = a a a Sec@ΦD I + Tan@ΦD M Φ âφ = à a Cos@ΦD â Φ a - Ha + a Cos@ ΦDL â Φ + Yx ] - Xx\ = 4 Sin@ ΦD - a - = a () = a () 3

16 Phi(k)_dk_5.nb Particle Wavefunction in k-space Using the wavefunction Ψ x, t A x a k x Ω t (a) Determine the momentum space wavefunction Φ k x and verify that it is normalized. (b) Determine the average value of k x using the momentum space wavefunction Φ k x. (c) Determine the uncertainty in momentum i.e. k x k x. (d) Does this wavefunction satisfies the uncertainty relationship x p x ħ (a) Determine the momentum space wavefunction Φ k x and verify that it is normalized. Previously we found that the normalization constant is A for convenience in this problem. The momentum wavefunction Φ k is determined by a φ where φ can be taken to be zero Φ k Ψ x, t k x x () where k p x ħ (for simplicity, we have dropped the subscript x). Φ k Ψ x, k x x x k a x a k x x () This is a difficult integral which can be done using contour integrals or by plugging into Mathematica Integrating Φ k Ψ x, k x x x k a x a k x x (3) a a k k To see whether Φ k x is normalized, we integrate the probability density, Φ k x k, from to. Φ k k a a k k k (4)

17 Phi(k)_dk_5.nb Making the variable change with Φ k k a a z z a a z z a a z z (5) a a z z (b) Determine the average value of k using the momentum space wavefunction Φ k x. To determine the average value of k, we multiply k by the probability density, Φ k x k, and integrate over all space ( to ). k k Φ k k a a k k k k (6) Making the variable change z k k and z k with z we obtain k a a z z k z (7) The integral over z vanishes by symmetry because the integrand is odd and we are integrating over a symmetric interval. The constant term is then our original normalization integral k k (8) This makes sense. The orginal wavefuction is multiplied by k x which is an eigenstate of momentum with momentum k. (c) Determine the uncertainty in momentum i.e. k k. As before, the above expression can be reduced to k k k k (9) To determine the average value of k, we multiply k by the probability density, Φ k x k, and integrate over all space ( to ). To simplify the integrals we again substitute k z k k k Φ k k a a k k k k () a a z z k z a a z z z k k z Performing the various integrals

18 Phi(k)_dk_5.nb 3 k a k () and so k k k k a k a () and p x k ħ ħ a (3) (d) Does this wavefunction satisfies the uncertainty relationship x p x ħ Previously we found that x a and we see then that these wavefunctions do satisfy the uncertainty relationship ħ x p x a ħ ħ a (4) (5)

19 GaussianRepresentationDeltaFunc.nb Gaussian Representation of a Dirac Delta Function Show by explicit integration that the following representation for the delta function does in fact agree with the definition of the delta function x - x lim ε ε ε - x-x Hint: Multiply the delta function by a function f x and expand the function in a Taylor series and integrate term by term. Then take the limit in the delta function definition. The Taylor expansion of f x about the point x is given by f x f x x x f x x x x x O x x 3 n f n x n n () Multiply by ε - x-x ε and integrate from to ε - x-x ε f x x ε ε n - x-x ε n n f n x x x n x n f n x x x n - x-x ε x () Change the integration variable to ε y x x and ε y x and ε - x-x ε f x x ε n n f n x ε n y n -y ε y n ε n n f n x y n -y y (3) Now the integral on the right is just a pure number. If n is odd then the integral is zero by symmetry. If n is even, then the integral can be related to the Gamma function. For n m even, the integral is just twice the integral from to. Leting z y and y z z z and so y m e -y y y m e -y y z m e -z z z z m e -z z m m m m (4) Therefore

20 GaussianRepresentationDeltaFunc.nb ε - x-x ε f x x m m f m m x m m ε m ε m f m x m m m In the limit as ε all terms in the sum vanish except for the terms with m which becomes lim ε ε - x-x ε f x x lim ε m ε m f m x m m (6) f x Operationally, it behaves just like the delta function x x

By using the operational definition of the delta function

the following operational statements: 8( x) = 8(x) 8(ax)= ö(x)

x) = ö (x) ö(f(x)) = ) where f(x 8(x x ) = Hint: Consider the

21 By using the operational definition of the delta function ff(x)s(x)dx = f(o) where f(x) is a well-behaved function, verify the following operational statements: 8( x) = 8(x) 8(ax)= ö(x) where ao xs(x)=o f(x)8 (x)= f (x)s(x) where 8 (x) _d[8(x)]/dx S ( x) = ö (x) ö(f(x)) = ) where f(x 8(x x ) = Hint: Consider the Taylor expansion of f(x) ö(x _a)=_j(ö(x_a)+s(x+a)) (o) (o).l. CLX) (L4 L L /) ± i LG C,,

22 () (b) Co &LL cc(x) (d) -flx = JCiC) ccoc) y X C) (is X5 I ). L.) $ ()C) = cck) Lx) d So j - rcni - O= 7?c4. S aj - t4l-kb -y -L ( IL) = - f() (c) r r j ç g) = so (,c) : L-cC) 3 cl x

23 a, ç, ((.. 7c)(f)I 4 x A ss.- i j o 44 I cj) f.çvc. ZIG I I - I c (.xz_o) S-CO -c Itc.çL).) g (_J clc)(+ (xic ajv.isq I. 4ltLCI Cc.i644 o cf vjl (xt-oc!x = = t (x-o.. Tv O.c X a. ) kt ic çcl4 oo Tyior,c -CA va,s.l&, o4. X C t..). ( 4 (.= SO &( 4cx] (XXe ) ) o.w.c so (cto)

Physics 217 Problem Set 1 Due: Friday, Aug 29th, 2008

Physics 217 Problem Set 1 Due: Friday, Aug 29th, 2008 Problem Set 1 Due: Friday, Aug 29th, 2008 Course page: http://www.physics.wustl.edu/~alford/p217/ Review of complex numbers. See appendix K of the textbook. 1. Consider complex numbers z = 1.5 + 0.5i and