Introduction to Quantum Mechanics 3rd. Edition by David J. Griffiths, Darrell F. Schroeter | 9781107189638 | 1107189632 | Holooly

Introduction to Quantum Mechanics [EXP-25631]

30 SOLVED PROBLEMS

Question: 2.6

A free particle, which is initially localized in the range , is released at time :ψ(x,0)=A, -a<x<a 0,otherwise where A and a are positive real constants. Find ψ(x,t)

Verified Answer:

First we need to normalize

\Psi (x,0) [/lat...

Question: 2.2

A particle in the infinite square well has the initial wave function Ψ(x,0)=Ax(a-x), (0≤x≤a), for some constant A (see Figure 2.3). Outside the well, of course,Ψ=0 . Find Ψ(x,t) .

Verified Answer:

First we need to determine A, by normalizing Ψ(x,...

Question: 3.4

A particle of mass m is bound in the delta function well V(x)=-αδ (x). What is the probability that a measurement of its momentum would yield a value greater than P0=m α/h ^2 ?

Verified Answer:

The (position space) wave function is (Equation 2....

Question: 6.7

A particle of mass m moves in one dimension in a harmonic-oscillator potential: H=p^2/2m +1/2 m ω^2 x^2 Find the position operator in the Heisenberg picture at time t.

Verified Answer:

Consider the action of

\hat{x}_H

o...

Question: 6.6

A particle of mass m moves in one dimension in a potential V(x): H=p^2/2m +V(x) Find the position operator in the Heisenberg picture for an infinitesimal time translation δ.

Verified Answer:

From Equation 6.71

[\hat{U}(t)=\exp\left[-\...

Question: 6.4

a) Find (r^2) for all four of the degenerate n=2 states of a hydrogen atom.

Verified Answer:

From Equation 6.47

[\left\langle \begin{mat...

Question: 2.3

Check that Equation 2.20 is satisfied, for the wave function in Example 2.2. If you measured the energy of a particle in this state, what is the most probable result? What is the expectation value of the energy?

Verified Answer:

The starting wave function (Figure 2.3) closely re...

Question: 7.2

Consider a particle of mass m in a two-dimensional oscillator potential H^0=p^2 /2m +1/2 mω ^2 (x^2+y^2) to which is added a perturbation H’=ε mω ^2 xy The unperturbed first-excited state (with E^0=2hω ) is two-fold degenerate, and one basis for those two degenerate states is ψa^0=ψ0(x)\ψ1(y)=√2/π

Verified Answer:

In terms of the rotated coordinates, the Hamiltoni...

Question: 6.3

Consider an eigenstate of a central potential Ψnlm with energy En. Use the fact that the Hamiltonian for a central potential commutes with any component of L , and therefore also with L+ and L-, to show Ψnlm±1 that are necessarily also eigenstates with the same energy as Ψnlm.

Verified Answer:

Since the Hamiltonian commutes with

\hat{L}...

Question: 3.1

Consider the operator Q≡id/dφ where ϕ is the usual polar coordinate in two dimensions. (This operator might arise in a physical context if we were studying the bead-on-a-ring; see Problem 2.46.) Is Q hermitian? Find its eigenfunctions and eigenvalues.

Verified Answer:

Here we are working with functions f(ϕ) on the fin...

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