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Chapter 2: 2.14P (page 51)

A particle is in the ground state of the harmonic oscillator with classical frequency ω, when suddenly the spring constant quadruples, so ω'=2ω, without initially changing the wave function (of course, Ψwill now evolve differently, because the Hamiltonian has changed). What is the probability that a measurement of the energy would still return the value ℏω2? What is the probability of getting ℏω?

Short Answer

Expert verified

The probability of getting 12ℏωis zero, and the probability of ℏωis 0.943.

Step by step solution

01

The given values of the question

The particle is in ground state and its classical frequency is ω. when the spring becomes entangled in a never-ending tangle ω'=2ω.

02

The probabilities of ℏω2 and ℏω

The new allowed energies are En'=(n+12)ℏω'=2(n+12)ω=ℏω,3ℏω,5ℏω,....

so the probability of getting12ℏω is zero.

The probability of getting ℏωis P0=c02, where c0=∫Ψ(x,0)ψ0'dxwith

Substitute the known values inP0=c02

Thus, the probability of ℏωis 0.943.

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Most popular questions from this chapter

Solve the time-independent Schr ̈odinger equation for a centered infinite square well with a delta-function barrier in the middle:

V(x)={αδ(x)for-a<x<+a∞for|x|≥a

Treat the even and odd wave functions separately. Don’t bother to normalize them. Find the allowed energies (graphically, if necessary). How do they compare with the corresponding energies in the absence of the delta function? Explain why the odd solutions are not affected by the delta function. Comment on the limiting cases α → 0 and α → ∞.

Show that E must be exceed the minimum value of V(x) ,for every normalizable solution to the time independent Schrodinger equation what is classical analog to this statement?

d2Ψdx2=2mh2[V(x)−E]Ψ;

IfE<Vmin thenΨ and its second derivative always have the same sign. Is it normalized?

a) Show that the wave function of a particle in the infinite square well returns to its original form after a quantum revival time T = 4ma2/π~. That is: Ψ (x, T) = Ψ (x, 0) for any state (not just a stationary state).


(b) What is the classical revival time, for a particle of energy E bouncing back and forth between the walls?


(c) For what energy are the two revival times equal?

A particle in the infinite square well has the initial wave function

ψ(X,0)={Ax,0≤x≤a2Aa-x,a2≤x≤a

(a) Sketch ψ(x,0), and determine the constant A

(b) Findψ(x,t)

(c) What is the probability that a measurement of the energy would yield the valueE1 ?

(d) Find the expectation value of the energy.

The transfer matrix. The S- matrix (Problem 2.52) tells you the outgoing amplitudes (B and F)in terms of the incoming amplitudes (A and G) -Equation 2.175For some purposes it is more convenient to work with the transfer matrix, M, which gives you the amplitudes to the right of the potential (F and G)in terms of those to the left (A and b):

(FG)=(M11M12M21M22)(AB)[2.178]

(a) Find the four elements of the M-matrix, in terms of the elements of theS-matrix, and vice versa. ExpressRI,TI,RrandTr(Equations 2.176and 2.177) in terms of elements of the M-matrix.,

(b) Suppose you have a potential consisting of two isolated pieces (Figure 2.23 ). Show that the M-matrix for the combination is the product of the twoM-matrices for each section separately: M=M2M1[2.179]

(This obviously generalizes to any number of pieces, and accounts for the usefulness of the M-matrix.)

FIGURE : A potential consisting of two isolated pieces (Problem 2.53 ).

(c) Construct the -matrix for scattering from a single delta-function potential at point V(x)=-αδ(x-a) :

(d) By the method of part , find the M-matrix for scattering from the double delta functionV(x)=-α[δ(x+a)+δ(X-a)] .What is the transmission coefficient for this potential?
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