Question

A particle of mass m is in the first excited state (i.e., n = 2) of an ISW that extends from 0 to a in the usual way. Suddenly, the well expands by 50%, becoming an ISW that extends from 0 to 3a/2. At the instant the wall moves, the wavefunction is not altered (although after the wall moves, the wavefunction will begin to evolve as dictated by the Schroedinger Equation with a new potential). Note: Having a suddenly expanding well is just another way of specifying € Ψ(x,0).

a) The energy of the particle is now measured. What is the most probable result? What is the probability of getting that result?

b) What is the second-most probable result? What is the probability of getting that result?

Answer #1

A particle is in the ground state of an infinite square well.
The potential wall at x = L suddenly (i.e., instantaneously) moves
to x = 3L. such that the well is now three times its original size.
(a) Let t = 0 be at the instant of the sudden change in the
potential well. What is ψ(x, 0)?
(b) If you measure the energy of the particle in the new well,
what are the possible energies?
(c) Estimate the...

Suppose initially a particle is in the ground state of a
1-dimensional inﬁnite square well which extends from x = 0 → a. The
wall of the square well is suddenly moved to 2a, so the square well
now extends from x = 0 → 2a. What is the probability of ﬁnding the
particle in the n = 3 state of the new (larger) square well?

7. A particle of mass m is described by the wave function ψ ( x)
= 2a^(3/2)*xe^(−ax) when x ≥ 0
0 when x < 0
(a) (2 pts) Verify that the normalization constant is correct.
(b) (3 pts) Sketch the wavefunction. Is it smooth at x = 0? (c) (2
pts) Find the particle’s most probable position. (d) (3 pts) What
is the probability that the particle would be found in the region
(0, 1/a)? 8. Refer to the...

For a particle in the first excited state of harmonic oscillator
potential,
a) Calculate 〈?〉1, 〈?〉1, 〈? 2〉1, 〈? 2〉1.
b) Calculate (∆?)1 and (∆?)1.
c) Check the uncertainty principle for this state.
d) Estimate the length of the interval about x=0 which
corresponds to the classically allowed domain for the first excited
state of harmonic oscillator.
e) Using the result of part (d), show that position uncertainty
you get in part (b) is comparable to the classical range of...

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