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Chapter 3: 3.9P (page 105)

(a) Cite a Hamiltonian from Chapter 2 (other than the harmonic oscillator) that has only a discrete spectrum.

(b) Cite a Hamiltonian from Chapter 2 (other than the free particle) that has only a continuous spectrum.

(c) Cite a Hamiltonian from Chapter 2 (other than the finite square well) that has both a discrete and a continuous part to its spectrum.

Short Answer

Expert verified

a) Infinite square well.

b) Finite rectangular barrier.

c) Finite square well.

Step by step solution

01

Example of a Hamiltonian that has a discrete spectrum

a)

The infinite square well Hamiltonian is an example of a Hamiltonian with a discrete spectrum. Recall that all energies Encorresponding to this Hamiltonian are discrete.

02

Example of a Hamiltonian that has a continuous spectrum

b)

The finite rectangular barrier Hamiltonian is an example of a Hamiltonian with a continuous spectrum. Recall that all energies corresponding to this Hamiltonian are continuous.

03

Example of a Hamiltonian that has both continuous and discrete spectrum

c)

The finite square well Hamiltonian is an example of a Hamiltonian with both a continuous and discrete spectrum. Recall that this Hamiltonian admits both bound (discrete spectrum) and scattering (continuous spectrum) status.

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Most popular questions from this chapter

(a) For a function f(x)that can be expanded in a Taylor series, show that f(x+x0)=eip^x0Ihf(x)

wherex_{0}

is any constant distance). For this reason, p^/his called the generator of translations in space. Note: The exponential of an operator is defined by the power series expansion: eQ^鈮�1+Q^+(1/2)Q^2+(1/3!)Q^3+...

(b) If 蠄(x,t)satisfies the (time-dependent) Schr枚dinger equation, show that 蠄(x,t+t0)=e-iH^t0/h蠄(x,t)

where t_{0}is any constant time); -H^/his called the generator of translations in time.

(c) Show that the expectation value of a dynamical variableQ(x,p,t), at time , t+t0can be written34

<Q>t+t0=<蠄x,t|eiH^t0/hQ^x^,p^,t+t0e-iH^t0/h|蠄x,t>

Use this to recover Equation 3.71. Hint: Lett0=dt, and expand to first order in dt.

(a) Write down the time-dependent "Schr枚dinger equation" in momentum space, for a free particle, and solve it. Answer: exp(-ip2t/2mh)桅(p,0).

(b) Find role="math" localid="1656051039815" 桅(p,0)for the traveling gaussian wave packet (Problem 2.43), and construct 桅(p,t)for this case. Also construct |桅(p,t)|2, and note that it is independent of time.

(c) Calculaterole="math" localid="1656051188971" pandrole="math" localid="1656051181044" p2by evaluating the appropriate integrals involving桅, and compare your answers to Problem 2.43.

(d) Show thatrole="math" localid="1656051421703" <H>=<p>2/2m+<H>0(where the subscript denotes the stationary gaussian), and comment on this result.

Show that the energy-time uncertainty principle reduces to the "your name" uncertainty principle (Problem 3.14), when the observable in question is x.

Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if P^and Q^have a complete set of common eigenfunctions, then[P^路Q^]f=0 for any function in Hilbert space.

Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if P^and Q^have a complete set of common eigenfunctions, then [P^.Q^]f=0for any function in Hilbert space.
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