Problem 1: The wavefunction for a particle is shown below. (a) What is the normalization constant A? Answer: 2 (b) What is the probability of finding the particle in the region -1/2 < x < 1/2? Answer: = 0.875 = 87.5% 0 if -1>x A(1+x) if -1≤x≤0 4 = A(1-x) 0 if 0≤x≤1 if x >1
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- Find the expected value of the continuous random variable X associated with the probability density function over the indicated interval. 3 f(x)=x²; [0,8] 512 E(X)= 8 XA particle confined in a one-dimensional box of length L(<= X <= L) is in a state described by the wave function where **check attached image** where A and B are constants given by real numbers. A) Determine which relationship A and B must satisfy for the wavefunction to be normalized. B) Suppose that A = B. What is the probability of the particle being found in the interval 0 <= X <= L/2? C) What are the values of A and B that minimize the probability of finding the particle in the range of positions 0 <= X <= L/2?QUESTION 1 The expectation value is the strict average of the possible values. O True False
- Calculate the uncertainties dr = V(x2) and op = V(p²) for %3D a particle confined in the region -a a, r<-a. %3DAn 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.3. A system of N harmonic oscillators of frequency w are prepared in identical initial states of wavefunction Þ(x,0). It is found that the measurement of the energy of the system at t = 0 gives 0.5hw with probability 0.25, 1.5hw with probability 0.5 and 2.5hw with probability 0.25. a. Write a possible function p(x, 0). b. Write the corresponding (x, t). c. What is the expectation value of the Hamiltonian in the state p(x,t) ? d. Calculate the expectation value of position at time t
- Consider a finite potential step with V = V0 in the region x < 0, and V = 0 in the region x > 0 (image). For particles with energy E > V0, and coming into the system from the left, what would be the wavefunction used to describe the “transmitted” particles and the wavefunction used to describe the “reflected” particles?A particle is confined to the region 0If the Probability of each of the steps of the drunk man is equal (p = q = 1/2) obtain the Probability that act of 5 steps 3 of them would be to the right. . Calculate the relative width for random walk problem. if the total number of steps is N = 25 a. for equal number of steps to the right and left side (p = q) . b. for p = 0.4 , q = 0.6Answer all these questionsAt t=0 a particle has the wave function -a4. a) Consider a square potential well which has an infinite barrier at x = 0 and a barrier of height U at x = L, as shown in the figure. For the case E L) that satisfy the appropriate boundary conditions at x = 0 and x = o. Put the appropriate conditions on x = L to find the allowed energies of the system. Are there conditions for which the solution is not possible? explain. U E L.SEE MORE QUESTIONSRecommended textbooks for youCollege PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage Learning
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