a) Find the potential V(x) associated with the wavefunction ψ(x) = Csech(ax) given that its energy eigenvalue is zero (i.e. E = 0).
b) Plot V(x) and ψ(x) on the same graph.
a) Find the potential V(x) associated with the wavefunction ψ(x) = Csech(ax) given that its energy...
5. The function x< 0 0 < x < a ψ(x)-Ax(1-(x/a)] is an acceptable wavefunction for a particle in a one-dimensional space (x can take values between -oo and +oo) (a) Give two reasons why this is an acceptable wave function. (b) Calculate the normalization constant A. (c) Using the definition for the average of an observable "o" described by the operator "o": and to)
Consider the following. Find the force F_x associated with the potential-energy function U = Ax^5, where A is a constant. (Use the following as necessary: A and x.) F_x = At what value(s) of x does the force equal zero? x =
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, being continuous everywhere. Sketch the wavefunction on your sketch of the potential energy.
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, being continuous everywhere. Sketch the wavefunction on your sketch of the potential energy.
A particle in the harmonic oscillator potential, V(x) - m2t2, is at time t 0 in the state ψ(x, t-0) = A3ψο(x) +4ψι (2)] where vn (z) is the nth normalized eigenfunction (a) Find A so that b is normalized. (b) Find ψ(x,t) and |ψ(x, t)12 (c) Find x (t) and p)(t). what would they be if we replaced ψ1 with V2? (hint: no difficult calculations are required) Check that Ehrenfest's theorem (B&J 3.93) holds for this wavefunction. (d) What...
mechani mie The potential energy barrier shown below is a simplified model of thec electrons in metals. The metal workfunction (Ew), the minimum energy required to remove an electron from the metal, is given by Ew-,-E where 1s the height of the potential energy barrier and E is the energy of the electrons near the surface of the metal. The potential energy barrier is = 5 eV V(x) V=0 (a) The wavefunction of an electron on the surface (x< 0)...
Question #9 all parts thanks 9. The wavefunction, p(x,t), of a particle moving along the x-axis, whose potential energy V(x) is independent of time, is described by the one-dimensional non-relativistic Schrödinger equation (where m is its mass, h is the reduced Planck constant, i is the imaginary number): 2m (a) Verify that it is a parabolic equation (page E-1-2). [It has wave-like solutions, however.] (b) Use the substitution Px,t)-Xx)Tt) to separate the equation into two ODEs. (c) Solve for T,...
4. A (one dimensional) particle in a box of length 2a (i.e., zero potential energy) is represented by the wavefunction v(x) 0, otherwise a. Sketch the wavefunction. Write down the (time independent) Schrodinger equation. Show whether or not the wavefunction is a solution to the equation. b. What does it mean physically if the wavefunction of the particle is NOT a solution to the Schrodinger equation? Explain. c. Determine the normalization constant A. 5. Same system. Find the average or...
8. Consider one electron in a 1D box of side L. Its wavefunction is given by V3 V3 2V3i where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, A, of a particle in a 1D box, h2 d2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is ų (x) an eigenfunction of A? If it is an eigenfunction, what is the eigenvalue?
Probability. A wavefunction ψ(x) describing the state of a particle free to move along one dimension x is given by the following expression: x <0,x>2L (A) Determine the value of the normalization constant c. (B) Draw the wavefunction. (C) Calculate Prob(L/2 S x 3 3L/2), the probability of finding the particle between x - L/2 and 3L/2 Probability. A wavefunction ψ(x) describing the state of a particle free to move along one dimension x is given by the following expression:...