(a)
The expectation value of
(a)
Answer to Problem 3.42P
The expectation value of
Explanation of Solution
Write the expression for the expectation value of the position.
Here,
Write the expression for the
Write the expression for the expectation value of momentum.
Write the expectation value of
Conclusion:
Therefore, the expectation value of
(b)
The value of
(b)
Answer to Problem 3.42P
The value of
Explanation of Solution
Write the expression for the
Use equation (I) and (II) to solve for
Write the expression for
Use equation (III) and (IV) to solve for
Use equation (VII) and (VIII) to find
Conclusion:
Therefore, the value of
(c)
Show that the expansion coefficients are
(c)
Answer to Problem 3.42P
It is showed that the expansion coefficients are
Explanation of Solution
Write the expression for the
Conclusion:
Therefore, it is showed that the expansion coefficients are
(d)
The value of
(d)
Answer to Problem 3.42P
The value of
Explanation of Solution
Write the expression for the normalization of
Conclusion:
Therefore, the value of
(e)
Show that
(e)
Answer to Problem 3.42P
It is showed that
Explanation of Solution
Write the expression for
Apart from the overall phase factor
Conclusion:
Therefore, it is showed that
(f)
The value of
(f)
Answer to Problem 3.42P
The value of
Explanation of Solution
It is given that the value of
Use equation (I) to solve for the value of
Conclusion:
Therefore, the value of
(g)
Whether the ground state
(g)
Answer to Problem 3.42P
Yes, the ground state
Explanation of Solution
Write the expression for the given value of
From equation (XIV), it is known that the ground state
Conclusion:
Therefore, the ground state
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Chapter 3 Solutions
Introduction To Quantum Mechanics
- 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])*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_forwardThe energy of the Hamiltonian operator defined below for the one-dimensional anharmonic oscillator Calculate first-order contributions to eigenvalues. (Here ? is a small number.)arrow_forwardTwo mass points of mass m1 and m2 are connected by a string passing through a hole in a smooth table so that m1 rests on the table surface and m2 hangs suspended. Assuming m2 moves only in a vertical line, what are the generalized coordinates for the system? Write the Lagrange equations for for the system and, if possible, discuss the physical significance any of them might have. Reduce the problem to a single second-order differential equation and obtain a first integral of the equation. What is its physical significance? (Consider the motion only until m1 reaches the hole.)arrow_forward
- According to Ehrenfest's theorem, the time evolution of an expectation value <A>(t) follows the Ehrenfest equations of motion (d/dt)<A>(t) = (i/[hbar])<[H,A]>(t). For the harmonic oscillator, the Hamiltonian is given by H = p2/2m + m[omega]2x2/2. a) Determine the Ehrenfest equations of motion for <x> and <p>. b) Solve these equations for the initial conditions <x> = x0, <p> = p0, where x0 and p0 are real constants. Better formatted version of the question attached.arrow_forwardA 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_forwardEvaluate the commutator [Â,B̂] of the following operators.arrow_forward
- Obtain the required relation pleasearrow_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_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_forward
- 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.)arrow_forwardCalculate the hermitian conjugate (adjoins) for operator d/dxarrow_forwardObtain the value of the Lagrange multiplier for the particle above the bowl given by x^2+y^2=azarrow_forward
- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning