Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 9.2, Problem 9.3P
To determine
The approximate transmission probability for a particle of energy
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The Hamiltonian of a spin in a constant magnetic field B aligned with the y axis is given by H = aSy, where a is a constant.
a) Use the energies and eigenstates for this case to determine the time evolution psi(t) of the state with initial condition psi(0) = (1/root(2))*matrix(1,1). (Vertical matrix, 2x1!)
b) For your solution from part (a), calculate the expectation values <Sx>, <Sy>, <Sz> as a function of time.
I have attached the image of the orginial question!
What does your result for the potential energy U(x=+L) become in the limit a→0?
At what displacements is the probability density a maximum for a state of a harmonic oscillator with v = 1? (Express your answers in terms of the coordinate y.)
Chapter 9 Solutions
Introduction To Quantum Mechanics
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- The wavefunction of is Ψ(x) = Axe(−ax^2)/2 for with energy E = 3aℏ2/2m. Find the bounding potential V(x). Looking at the potential’s form, can you write down the two energy levels that are immediately above E?arrow_forwardThe Hamiltonian of a spin in a constant magnetic field B aligned with the y axis is given by H = aSy, where a is a constant. a) Use the energies and eigenstates for this case to determine the time evolution [psi](t) of the state with initial condition [psi](0) = (1/root[2])*mat([1],[1]). b) For your solution from part (a), calculate the expectation values <Sx>, <Sy>, <Sz> as a function of time. Better formatted version of the question is attached.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_forward
- The wavefunction of is Ψ(x) = Axe−αx2/2 for with energy E = 3αℏ2/2m. Find the bounding potential V(x). Looking at the potential’s form, can you write down the two energy levels that are immediately above ??arrow_forward1. a. For a free particle, write the relations between the wave vector k and itsmomentum vector p and angular frequency ω and its energy E.b. What is the general form in one dimension of the wave function for a freeparticle of mass m and momentum p?c. Can this wave function ever be entirely real? If so, show how this ispossible. If not, explain why not.d. What can you say about the integral of the |Ψ (x; t)|^2 from - ∞ to + ∞ ?e. Is this a possible wave function for a real, physical particle? Explain whyor why not.arrow_forwardCalculate the expectation value of x2 in the state described by ψ = e -bx, where b is a ħ constant. In this system x ranges from 0 to ∞.arrow_forward
- A point particle moves in space under the influence of a force derivablefrom a generalized potential of the formU(r, v) = V (r) + σ · L,where r is the radius vector from a fixed point, L is the angular momentumabout that point, and σ is the fixed vector in space. Find the components of the force on the particle in spherical polar coordinates, on the basis of the equation for the components of the generalized force Qj: Qj = −∂U/∂qj + d/dt (∂U/∂q˙j)arrow_forwardUse the fact that at the critical point the first and second partial derivatives of P with respect to Vm at constant T are zero (∂P/∂Vm=∂2P/∂V2m=0) to derive the expressions for the Van der Waals constants in terms of critical parameters. Show full and complete procedure, do not skip any steparrow_forwardFor a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is. 1 eV less than μarrow_forward
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