The wave function of a particle at t = 0 is given as: ψ(x, t) = C exp[ -|x|/x0] where C and x0 are constants. (a) What is the relation between C and x0? (b) Calculate the expectation value of position x of the particle. (c) Suggest a region in x in which the probability of finding the particle is 0.5.

The wave function of a particle at t = 0 is given as: ψ(x, t) = C exp[ -|x|/x0] where C and x0 are constants. (a) What is the relation between C and x0? (b) Calculate the expectation value of position x of the particle. (c) Suggest a region in x in which the probability of finding the particle is 0.5.

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter28: Quantum Physics

Section: Chapter Questions

Problem 56P

See similar textbooks

Similar questions

Suppose a wave function is discontinuous at some point. Can this function represent a quantum state of some physical particle? Why? Why not?
In a simple model for a radioactive nucleus, an alpha particle (m = 6.64 * 10-27 kg) is trapped by a square barrier that has width 2.0 fm and height 30.0 MeV. (a) What is the tunneling probability when the alpha particle encounters the barrier if its kinetic energy is 1.0 MeV below the top of the barrier (Fig. )? (b) What is the tunneling probability if the energy of the alpha particle is 10.0 MeV below the top of the barrier?
(a) Use the standard definition of the average value of a random variable given its probability density function: = 10 x p₁(x) -∞ (9) to find average position of the wave-particle for each stationary state. Recall that pn = Un is the probability density associated with the stationary state n = 1,2,... for the particle in a box. Also, n(x, t) is only defined for x = (0, L). Hint: the result is the same for all stationary states. That is, it does not depend on n. n= dx.
The wavefunction of an electron is given by 1≤x≤ 16 y(x) = c = 0 x 16 where c is a positive real number. (a) Calculate the probability of finding the electron in the range 2 ≤ x ≤ 11. (b) Where is the electron most likely to be found in the range 1 ≤ x ≤ 16? Explain your answer briefly, with one or two sentences only.
(WF-1) The wave function for an electron moving in ID is given by: y(x) = C(x - ix²) for 0 ≤ x ≤ 1 and zero everywhere else. (a) Find the value of the normalization constant (C) for the wave function. (b) What is the probability of locating the electron between x = 0 and x = 1/4? (c) What is the probability of locating the electron between x = 1/4 and x = 3/4?
At a certain instant of time, a particle has the wave function y (x) A xe(- |x|/B where b — 3 пт: а) Normalize the function. If at that instant a measurement of x is made, find the probability that the result: b) is between 0.9000 and 0.9001 nm, is between O and 2 nm. c) For what value of x is the probability density minimal?
A particle is in a three-dimensional box. The y length of the box is twice the x length, and the z length is one-third of the y length. (a) What is the energy difference between the first excited level and the ground level? (b) Is the first excited level degenerate? (c) In terms of the x length, where is the probability distribution the greatest in the lowest-energy level?
An electron has a wave function Y(x) = Ce-kl/xo %3D where x0 is a constant and C = 1//x0 is the normalization constant. In this case, get the expressions for (x) And Ax in terms of x0. Also calculate the probability that the electron is found within the standard deviation of its average position, that is, in the range (x) –Ax a (x) + Ax, show that this is independent of x0.
The wave function of a certain particle is y= A cos²x for -t/2 < x
A particle is confined between rigid walls separated by a distance L = 0.189 nm. The particle is in the second excited state (n = 3). Evaluate the probability to find the particle in an interval of width 1.00 pm located at (a) x=0.188 nm; (b) x = 0.031 nm; (c) x = 0.079 nm. (Hint: No integrations are required for this problem; use Eq. 5.7 directly.) (d) What would be the corresponding results for a classical particle?
Calculate the transmission probability for quantum-mechanical tunneling in each of the following cases. (a) An electron with an energy deficit of U - E= 0.010 0 eV is incident on a square barrier of width L = 0.100 nm. (b) An electron with an energy deficit of 1.00 eV is incident on the same barrier. (c) An alpha particle (mass 6.64 × 10-27 kg) with an energy deficit of 1.00 MeV is incident on a square barrier of width 1.00 fm. (d) An 8.00-kg bowling ball withan energy deficit of 1.00 J is incident on a square barrier of width 2.00 cm.
Consider a particle with the following wave-function: ,xL and L and A are constants. (a) What is the normalization constant A (in terms of L)? (b) What is (in terms of L)? ? (ie, ) (c) What is the probability that the particle will be found within x=0 and L/2?