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Chapter 2: Q26P (page 77)
What is the Fourier transform ? Using Plancherel鈥檚 theorem shows that.
The Fourier transform for the given function is
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In this problem we explore some of the more useful theorems (stated without proof) involving Hermite polynomials.
a. The Rodrigues formula says that
Use it to derive and .
b. The following recursion relation gives you in terms of the two preceding Hermite polynomials:
Use it, together with your answer in (a), to obtain and .
(c) If you differentiate an nth-order polynomial, you get a polynomial of
Order (n-1). For the Hermite polynomials, in fact,
Check this, by differentiatingand .
d. is the nth z-derivative, at z = 0, of the generating function or, to put it another way, it is the coefficient of in the Taylor series expansion for this function:
Use this to obtain and .
A particle of mass m in the harmonic oscillator potential (Equation 2.44) starts out in the state for some constant A.
(a) What is the expectation value of the energy?
(c) At a later time T the wave function islocalid="1658123604154"
for some constant B. What is the smallest possible value of T ?
Delta functions live under integral signs, and two expressions involving delta functions are said to be equal if
for every (ordinary) function f(x).
(a) Show that
(2.145)
where c is a real constant. (Be sure to check the case where c is negative.)
(b) Let be the step function:
(2.146).
(In the rare case where it actually matters, we define to be 1/2.) Show that
A particle is in the ground state of the harmonic oscillator with classical frequency , when suddenly the spring constant quadruples, so , without initially changing the wave function (of course, will now evolve differently, because the Hamiltonian has changed). What is the probability that a measurement of the energy would still return the value ? What is the probability of getting ?
Prove the following three theorem;
a) For normalizable solutions the separation constant E must be real as and show that if equation 1.20 is to hold for all must be zero.
b) The time - independent wave function localid="1658117146660" can always be taken to be real, This doesn鈥檛 mean that every solution to the time-independent Schrodinger equation is real; what it says is that if you鈥檝e got one that is not, it can always be expressed as a linear combination of solutions that are . So, you might as well stick to 鈥檚 that are real
c) If is an even function then can always be taken to be either even or odd
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