Quantum Mechanics: Concepts and Applications 2nd. Edition by Nouredine Zettili | 9780470746561 | 0470026790 | Holooly

Quantum Mechanics: Concepts and Applications [EXP-27107]

216 SOLVED PROBLEMS

Question: 3.P.5

(a) Assuming that the system of Problem 3.4 is initially in the state |Φ3〉, what values for the energy and the observable A will be obtained if we measure: (i)H first then A, (ii) A first then H? (b) Compare the results obtained in (i) and (ii) and infer whether H and A re compatible. Calculate [H

Verified Answer:

(a) (i) The measurement of H first then A is repre...

Question: 4.P.14

(a) Assuming the potential seen by a neutron in a nucleus to be schematically represented by a one-dimensional, infinite rigid walls potential of length 10 fm, estimate the minimum kinetic energy of the neutron. (b) Estimate the minimum kinetic energy of an electron bound within the nucleus

Verified Answer:

The energy of a particle of mass m in a one-dimens...

Question: 5.1

(a) Calculate the commutators [^X, ^Lx],[^X, ^Ly], and [^X, ^Lz]. (b) Calculate the commutators: [^Px, ^Lx],[^Px, Ly], and [^Px, ^Lz]. (c) Use the results of (a) and (b) to calculate [^X, ^→L^2] and [^Px, ^→L^2].

Verified Answer:

(a) The only nonzero commutator which involves [la...

Question: 11.P.1

(a) Calculate the differential cross section in the Born approximation for the potential V(r) = V0 e^-r/R /r, known as the Yukawa potential. (b) Calculate the total cross section. (c) Find the relation between V0 and R so that the Born approximation is valid.

Verified Answer:

(a) Inserting

V(r)=V_{0} e^{-r / R} / r[/la...

Question: 11.2

(a) Calculate the differential cross section in the first Born approximation for a Coulomb potential V(r) = Z1Z2e^2/r , where Z1e and Z1e are the charges of the projectile and target particles, respectively. (b) To have a quantitative idea about the cross section derived in (a), consider the

Verified Answer:

In the case of a Coulomb potential,

V(r)=Z_...

Question: 6.P.7

(a) Calculate the expectation value 〈r〉21 for the hydrogen atom and compare it with the value r at which the radial probability density reaches its maximum for the state n = 2, l = 1. (b) Calculate the width of the probability density distribution for r.

Verified Answer:

(a) Since

R_{21} (r)=re^{-r/2a_{0} } /\sqrt...

Question: 4.P.15

(a) Calculate the expectation value of the operator X^4 in the N-representation with respect to the state |n〉 (i.e., 〈n | X^4| n〉). (b) Use the result of (a) to calculate the energy En for a particle whose Hamiltonian is H = P^2 /(2m) + 1/2 m ω^2 X^2 – λ X^4.

Verified Answer:

(a) Since

\Sigma ^{\infty }_{m=0}|m〉〈m|=1...

Question: 7.P.7

(a) Calculate the expression of 〈2, 0|Y10|1, 0 〉. (b) Use the result of (a) along with the Wigner–Eckart theorem to calculate the reduced matrix element 〈2 ||Y1 ||1 〉.

Verified Answer:

(a) Since

〈2,0|Y_{10}|1,0 〉=\int_{0}^{\pi...

Question: 1.P.15

(a) Calculate the final size of the wave packet representing a free particle after traveling a distance of 100 m for the following four cases where the particle is (i) a 25 eV electron whose wave packet has an initial width of 10^-6 m, (ii) a 25 eV electron whose wave packet has an initial width

Verified Answer:

(a) If the initial width of the wave packet of the...

Question: 3.P.12

(a) Calculate the Poisson bracket between the x and y components of the classical orbital angular momentum. (b) Calculate the commutator between the x and y components of the orbital angular momentum operator. (c) Compare the results obtained in (a) and (b).

Verified Answer:

(a) Using the definition (3.113)

\left\{A,B...

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