Explanation The wave function in region ψ x < 0 is the combination of incident particle’s wave function ψ i n c and reflected particle’s wave function ψ r e f . ψ x < 0 = ψ i n c + ψ r e f In order to find wave function in the region, the Schrödinger equation should be solved using the potential energy for that region. d ψ 2 d x 2 = − 2 m ℏ 2 ( E − U ( x < 0 ) ) ψ x < 0 = − 2 m ℏ 2 ( E − 0 ) ψ x < 0 = − 2 m E ℏ 2 ψ x < 0 The coefficients of wave function ψ x < 0 are grouped into variable k . k = 2 m E ℏ 2 Substitute values of variable k in Schrödinger equation. d ψ 2 d x 2 = − 2 m E ℏ 2 ψ x < 0 = − k 2 ψ x < 0 The solution to the Schrödinger equation in the region is the combination of positive and negative imaginary exponentials. ψ x < 0 = A e i k x + B e − i k x In order to find wave function in region, the Schrödinger equation should be solved using the potential energy for that region. d ψ 2 d x 2 = − 2 m ℏ 2 ( E − U 0 ) ψ 0 < x < L The coefficients of wave function ψ x < 0 are grouped into variable k ' . k ' = 2 m ( E − U 0 ) ℏ 2 Substitute the value of variable k ' in Schrödinger equation. d ψ 2 d x 2 = − 2 m ( E − U 0 ) ℏ 2 ψ 0 < x < L = − k ' 2 ψ 0 < x < L The solution to the Schrödinger equation in the region before the step is a combination of positive and negative imaginary exponentials. ψ x < 0 = C e i k ' x + D e − i k ' x To determine the wave function after the step, solve the Schrödinger equation, using the potential energy in that region 0 . d ψ 2 d x 2 = − 2 m ℏ 2 ( E − U ( x > L ) ) ψ x > L = − 2 m ℏ 2 ( E − 0 ) ψ x > L = − 2 m E ℏ ψ x > L The coefficients of the wave function ψ x > L are the same as the coefficients in the region before the step. Regroup them into k 2 . d ψ 2 d x 2 = − 2 m E ℏ 2 ψ x > L = − k 2 ψ x > L The solution to me Schrödinger equation in this region is a combination of positive and negative exponentials. ψ x > L = F e i k x + G e − i k x There is no reason to have a wave travelling in the negative direction in this region, so the second coefficient G must be 0 . This simplifies the resulting wave. ψ x > L = F e i k x In order to be a valid solution, the wave functions on both sides of the step up must meet in the middle, meaning their limits must be the same. lim x → 0 ( ψ x < 0 ) = lim x → 0 ( ψ 0 < x < L ) A + B = C + D The other condition defining a valid solution isthe derivatives of the wave functions on each side of the step up must approach the same value. lim x → 0 ( d ψ x < 0 d x ) = lim x → 0 ( d ψ 0 < x < L d x ) k A − k B = k ' C − k ' D On the second side of the step, the same continuity conditions apply. The limits of the wave functions must be the same from both sides. lim x → L ( ψ 0 < x < L ) = lim x → L ( ψ x > L ) C e i k ' L + D e − i k ' L = F e i k L The limits of the derivatives must be the same as well

2nd Edition

ISBN: 9780805303087

Author: Randy Harris

Publisher: Addison Wesley

See similar textbooks

Concept explainers

Connection formulae for WKB approximation

In quantum mechanics, Schrödinger’s equation for certain systems with varying parameters cannot be solved directly. In such a case, an approximation method is used to obtain the approximate solution. WKB method is a special technique used to obtain the a…

Question

Chapter 6, Problem 22E

To determine

To Show: B=0

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6, Problem 21E

Chapter 6, Problem 23E

Students have asked these similar questions

It 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.

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?

V,E -V. Consider the potential well of depth V, and width w, as shown in the figure above. An electron with energy Ɛ > 0 is incident from the left. (a) Determine the condition involving the incident energy (E) and the depth (V.) and width (w) of the well for resonant transmission in which there is no reflected wave associated with the electron. (b) If 4(x) = Aejkx is the incident electron wave for x w) the well, under resonant conditions?

Chapter 6 Solutions

Modern Physics

Ch. 6 - Prob. 1CQ Ch. 6 - Prob. 2CQ Ch. 6 - Prob. 3CQ Ch. 6 - Prob. 4CQ Ch. 6 - Prob. 5CQ Ch. 6 - Prob. 6CQ Ch. 6 - Prob. 7CQ Ch. 6 - Prob. 8CQ Ch. 6 - Prob. 9CQ Ch. 6 - Prob. 10CQ

Ch. 6 - The diagram below plots (k) versus wave number for...Ch. 6 - Prob. 12CQ Ch. 6 - Prob. 13E Ch. 6 - Prob. 14E Ch. 6 - Prob. 15E Ch. 6 - Prob. 16E Ch. 6 - Prob. 17E Ch. 6 - Prob. 18E Ch. 6 - Prob. 19E Ch. 6 - Prob. 20E Ch. 6 - Prob. 21E Ch. 6 - Prob. 22E Ch. 6 - Prob. 23E Ch. 6 - Prob. 24E Ch. 6 - Prob. 25E Ch. 6 - Prob. 26E Ch. 6 - Prob. 27E Ch. 6 - Prob. 28E Ch. 6 - Obtain the smoothness conditions at the...Ch. 6 - Prob. 30E Ch. 6 - Prob. 31E Ch. 6 - Jump to Jupiter The gravitational potential energy...Ch. 6 - Prob. 33E Ch. 6 - Obtain equation (618) from (616) and (617).Ch. 6 - Prob. 35E Ch. 6 - Prob. 36E Ch. 6 - Prob. 37E Ch. 6 - Prob. 38E Ch. 6 - Prob. 39E Ch. 6 - Prob. 40E Ch. 6 - Prob. 41E Ch. 6 - Prob. 42E Ch. 6 - Prob. 43E Ch. 6 - Prob. 44E Ch. 6 - Prob. 45E Ch. 6 - Prob. 46E Ch. 6 - Prob. 47E Ch. 6 - Prob. 48E Ch. 6 - Prob. 49E Ch. 6 - Prob. 50E Ch. 6 - Prob. 51CE Ch. 6 - Prob. 52CE Ch. 6 - Prob. 53CE Ch. 6 - Prob. 54CE Ch. 6 - Prob. 56CE

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.

B = 0

Solution Summary: The author explains that the Schrödinger equation should be solved using the potential energy for the region.

Concept explainers

To Show: B=0

Want to see the full answer?

Chapter 6 Solutions