Modern Physics, 3rd Edition
3rd Edition
ISBN: 9780534493394
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
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Question
Chapter 7, Problem 6Q
To determine
The suitability of portraying wave functions and potential barriers on the same graph and discuss whether a wave whose crest falls below the top of square barrier ever penetrate the barrier.
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Determine the normalization constant for the wavefunction for a 3-dimensional box (3 separate infinite 1-dimensional wells) of lengths a (x direction), b (y direction), and c (z direction).
If the walls of a potential energy well have finite height, V0, the wave function that describes the particle inside the box does not vanish on the walls, but continues through the barrier power. Discuss, only conceptually, the diagrams drawn below for the three first energy levels. It is not to make a treatise about it, especially copied from books. It's simply to explain how much you understand about the concepts linked to graphics below.
A particle is confined to a 1D box between x=0 and x-1 and has the normalized wavefunction of
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difference in this context means that you should enter your final result as a positive number).
Chapter 7 Solutions
Modern Physics, 3rd Edition
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- Suppose a particle has zero potential energy for x < 0. a constant value V. for 0 ≤ x ≤ L. and then zero for x > L. Sketch the potential. Now suppose that wavefunction is a sine wave on the left of the barrier. declines exponentially inside the barrier. and then becomes a sine wave on the right. beingcontinuous everywhere. Sketch the wavefunction on your sketch of the potential energy.arrow_forwardcan you explain further, inside a finite well, the wave function is either cosine or sine, so exactly what wave function are we considering around x=0 to decide parity if we dont even know what wave we are using at this pointarrow_forwardA particle with mass m is in the lowest (ground) state of the infinte potential energy well, as defined in the provided image. At time t=0, the wall located at x = L is suddenly pulled back to a position at x = 2L. This change occurs so quickly that instantaneously the wave function does not change. Calculate the probability that a measurement of the energy will yield the ground-state energy of the new well. What is the probability that a measurement of the energy will yield the first excited energy of the new well?arrow_forward
- Enumerate the constraints that restrict which functions can be viable wavefunctions.arrow_forward10. Sketch the first four wavefunctions (n = 1 through n = 4) for the particle in a box, and sketch the probability densities for the same wavefunctions. Can you determine (x) for these wavefunctions without evaluating any integrals? Explain. Is this the same as the most likely location(s) for the particle, in each case?arrow_forwardInfinite/finite Potential Well 1. Sketch the solution (Wave function - Y) for the infinite potential well and show the following: (a) Specify the boundary conditions for region I, region II, and region III. (i.e. U = ?, and x = ?) (b) Specify the length of the potential well (L=10 cm) (c) Which region will have the highest probability of finding the particle?arrow_forward
- Why don't you include the time dependent part of the wave equation when finding the expectation value of the potential? I don't see how the modulus of the wave function gets rid of the time dependent part. Can you please explain?arrow_forwardFor a particle in a box of length L sketch the wavefunction corresponding to the state with the lowest energy and on the same graph sketch the corresponding probability density. Without evaluating any integrals, explain why the expectation value of x is equal to L/2.arrow_forwardA. at pattern. Let's ssuming that the e angular spatial poral frequencies w, correspond to e is then W, 1) ir) [7.33] avelength of the quals the group eing amplitude- e waves of fre- of modulating and sum over is called the wer sideband. 7.19 Given the dispersion relation w = ak', compute both the phase and group velocities. -7.20* Using the relation 1/v = dk/dv. prove that 1 Vg I V₂ 7.21* In the case of lightwaves, show that V = 1 Ve v dv v² dv V 7.22 The speed of propagation of a surface wave in a liquid of depth much greater than λ is given by v dn c dv 11 -+- C solve 7.25 only where g = acceleration of gravity, λ = wavelength, p = density. Y = surface tension. Compute the group velocity of a pulse in the long wavelength limit (these are called gravity waves). 7.23* Show that the group velocity can be written as dv dλ 7.24 Show that the group velocity can be written as gλ 2πΥ + 2πT pλ Vg = V-A ( n + w(dn/dw) 7.25 With the previous problem in mind prove that dn (v) dv n₂ = n(v) + v.…arrow_forward
- A half-infinite well has an infinitely high wall at the origin and one of finite height U_0 at x = L. The number of allowed states is limited just like the finite well. Making an assumption that it has only two energy states, E1 and E2, where E2 is not much below U_0.a. Make a sketch of the potential energy, then add plausible sketches of the two allowed wave functions on separate horizontal axes whose heights are E1 and E2.arrow_forwardConsider a particle in a box of length L with one end coinciding with the origin. Consider the state with n=2. 1. Compute the expectation value of position as a function of time 2. As well as the extrema in the probability density as a function of time. Interpret.arrow_forwardFor a particle in a square box of side L, at what position (or positions) is the probability density a maximum if the wavefunction has n1 = 1, n2 = 3? Also, describe the position of any node or nodes in the wavefunction.arrow_forward
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