Modern Physics
2nd Edition
ISBN: 9780805303087
Author: Randy Harris
Publisher: Addison Wesley
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Question
Chapter 6, Problem 19E
(a)
To determine
To show: The probability density to the left of the step is
(b)
To determine
The angle and the direction coefficient of the transmitted wave.
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Check out a sample textbook solutionStudents have asked these similar questions
Consider a beam of particles of kinetic energy E incident on a potential step at x = 0
that is U high, such that E > U, as illustrated in the accompanying figure.
Energy
E
Region I
x = 0
E-U
Region II
|²
1+r
Derive the equation for the reflection probability, R = |^|2 = (+7) 2.
where
KII
k1
r , where A+ and A_ are the amplitudes of the incoming and reflected waves,
k
respectively, and K is the wavenumber in the specified region.|
1 W:0E
Problem 1.17 A particle is represented (at time t = 0) by the wave function
| A(a? – x²). if -a < x < +a.
0,
Y (x, 0) =
otherwise.
(a) Determine the normalization constant A.
(b) What is the expectation value of x (at time t =
0)?
(c) What is the expectation value of p (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 p?.
(f) Find the uncertainty in x (ox).
In a scattering experiment, the potential is spherically symmetric and the particles are scat-
tered at such energy that only s and p waves need be considered. If the differential cross-section o (0)
can be written in the form o (0) = a+bcos0+ccos? 0 , then what are the values of a, b, c in the terms of
phase shifts? What is the value oftotal cross-section in terms ofa, b, c?
Chapter 6 Solutions
Modern Physics
Ch. 6 - Prob. 1CQCh. 6 - Prob. 2CQCh. 6 - Prob. 3CQCh. 6 - Prob. 4CQCh. 6 - Prob. 5CQCh. 6 - Prob. 6CQCh. 6 - Prob. 7CQCh. 6 - Prob. 8CQCh. 6 - Prob. 9CQCh. 6 - Prob. 10CQ
Ch. 6 - The diagram below plots (k) versus wave number for...Ch. 6 - Prob. 12CQCh. 6 - Prob. 13ECh. 6 - Prob. 14ECh. 6 - Prob. 15ECh. 6 - Prob. 16ECh. 6 - Prob. 17ECh. 6 - Prob. 18ECh. 6 - Prob. 19ECh. 6 - Prob. 20ECh. 6 - Prob. 21ECh. 6 - Prob. 22ECh. 6 - Prob. 23ECh. 6 - Prob. 24ECh. 6 - Prob. 25ECh. 6 - Prob. 26ECh. 6 - Prob. 27ECh. 6 - Prob. 28ECh. 6 - Obtain the smoothness conditions at the...Ch. 6 - Prob. 30ECh. 6 - Prob. 31ECh. 6 - Jump to Jupiter The gravitational potential energy...Ch. 6 - Prob. 33ECh. 6 - Obtain equation (618) from (616) and (617).Ch. 6 - Prob. 35ECh. 6 - Prob. 36ECh. 6 - Prob. 37ECh. 6 - Prob. 38ECh. 6 - Prob. 39ECh. 6 - Prob. 40ECh. 6 - Prob. 41ECh. 6 - Prob. 42ECh. 6 - Prob. 43ECh. 6 - Prob. 44ECh. 6 - Prob. 45ECh. 6 - Prob. 46ECh. 6 - Prob. 47ECh. 6 - Prob. 48ECh. 6 - Prob. 49ECh. 6 - Prob. 50ECh. 6 - Prob. 51CECh. 6 - Prob. 52CECh. 6 - Prob. 53CECh. 6 - Prob. 54CECh. 6 - Prob. 56CE
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- What would happen to the incident wave Þ(x) = Aelk* at x < 0 that sees a step potential in the form V(x) = iT in the positive x region? Here i is complex number and I is positive real number. (a) Explain in detail (perform some calculations too). (b) Calculate the reflection and transmission coefficients.arrow_forwardLet a⪯b⪯c⪯da⪯b⪯c⪯d be the variable ordering.ϕ=ϕ= a&b&d&!c|a&c&d|d&!b&!c|!dβ=β= a&b&c|!c a) Convert the formula ϕϕ to Shannon normal form. b) Convert the formula ββ to Shannon normal form. c) ψψ is obtained by replacing all occurences of the variable b by formula ββ in formula ϕϕ.Compute the ROBDD of ψψ by the Compose algorithm, and convert the result to Shannon normal form.arrow_forwardAn electron with an initial kinetic energy of 1.542 eV (in a region with 1.095 eV potential energy) is incident on a potential step (extending from x=0 to ∞) to V=2.381 eV. What is the transmission probability (in %)? FYI: If we had a travelling wave arriving at a similar potential DROP, then k1 (for x<0) would be real and the symmetry of R=(k1-k2)2/(k1+k2)2 implies reflection/transmission are the same as a potential RISE with the same energies but k1 and k2 swapped.arrow_forward
- Problem 1. Using the WKB approximation, calculate the energy eigenvalues En of a quantum- mechanical particle with mass m and potential energy V (x) = V₁ (x/x)*, where V > 0, Express En as a function of n; determine the dimensionless numeric coefficient that emerges in this expression.arrow_forwardIt is stated without proof with respect to Bragg’s law that when the atoms are not sym- metrically disposed to the incident and reflected beams (Fig. 8.3(b)), the path difference (AB + BC) = 2dhkl sin θ . Prove, using very simple geometry, that this is indeed the case.arrow_forwardConsider the wave function for the ground state harmonic oscillator: m w1/4 e-m w x2/(2 h) A. What is the quantum number for this ground state? v = 0 B. Enter the integrand you'd need to evaluate (x) for the ground state harmonic oscillator wave 'function: (x) = |- то dx e C. Evaluate the integral in part B. What do you obtain for the average displacement? 0arrow_forward
- Consider the following three wave functions: wi(y) = Aje¬", w2(v) = Aze-³/2, w3(y) = A3(e¬y* + ye¬³/2), where A1, A2, and A3 are normalization constants. (a) Find the constants A1, A2, and A3 so that w1, w2, and w3 are normalized. (b) Find the probability that each one of the states will be in the interval –1 < y < 1.arrow_forwardProblem # 2. In the two-level system, estimate the emission line full width at half maximum (FWHM) for spontaneous emission at 650 nm if the spontaneous radiative lifetime of the upper state is about 3,000 nanoseconds.arrow_forwarda. Calculate the minimum uncertainty in the position of an electron in meters, if its velocity has been measured to within 2.34e-06 m/s. For the mass of an electron use 9.10938356 × 10-31 kg and for huse 1.0545718 x 10-34 Js. m b. 44 = 141² = ² represents a: O the pitchfork operator O probability density O numerov program O wave function c. The sum of all the little bits of probability over possible measurements that can be made (i.e. over all space): S4²4 = 1 is called: O the quantum condition O the normalization condition O the human condition O the eignen conditionarrow_forward
- Problem 3 For = [-1,2] and X(@) = w? - 1, find expressions for Fx(x) and fx(x) assuming that probability measure on 2 is the Lebesgue measure scaled by 1/3.arrow_forwardA ID harmonic oscillator of angular frequency w and charge q is in its ground state at time t=0. A perturbation H'(t) = qE eA3 (where E is ekctric field and ß is a constant) is %3D applied for a time t = t. Cakulate the probability of transition to the first and second excited state. (hint: you may expand exponential in perturbation and keep it only up to linear term)arrow_forwardSuppose Fuzzy, a quantum-mechanical duck, lives in a world in which h = 2 J s. Fuzzy has a mass of 1.90 kg and is initially known to be within a pond 1.00 m wide. (a) What is the minimum uncertainty in the duck's speed? m/s (b) Assuming this uncertainty in speed to prevail for 4.90 s, determine the uncertainty in Fuzzy's position after this time. marrow_forward
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