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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition · 469 pages

ISBN: 9780131118928

Chapter 2: Q26P (page 77)

What is the Fourier transform $δ (x)$ ? Using Plancherel’s theorem shows that $δ (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{ikx} dk$ .

Short Answer

Expert verified

The Fourier transform for the given function is

$δ (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{ikx} dk$

Step by step solution

01

Define the Schrodinger equation

A differential equation that describes matter in quantum mechanics in terms of the wave-like properties of particles in a field. Its answer is related to a particle's probability density in space and time.

02

Determine the Fourier transform

By using the Fourier transform,

$F (k) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} f (x) e^{ikx} dx$

Substitute $f (x) = δ (x)$ into equation

$F (k) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} δ (x) e^{ikx} dx$

$\int_{- \infty}^{\infty} δ (x) e^{ikx} dx = 1$

$F (k) = \frac{1}{\sqrt{2} π}$

$\begin{matrix} f (x) = δ (x) \\ = \frac{1}{\sqrt{2} π} \int_{- \infty}^{\infty} \frac{1}{\sqrt{2} π} e^{ikx} dk \\ = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{ikx} dk \end{matrix}$

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Most popular questions from this chapter

Question: Find the probability current, $J$ (Problem 1.14) for the free particle wave function Equation $2.94$ . Which direction does the probability flow?

A particle in the harmonic oscillator potential starts out in the state $Ψ (x, 0) = A [3 ψ_{0} (x) + 4 ψ_{1} (x)]$

a) Find $A$ .

b) Construct $Ψ (x, t)$ and $| Ψ (x, t)^{2} |$

c) Find $⟨ x ⟩$ and $⟨ p ⟩$ . Don't get too excited if they oscillate at the classical frequency; what would it have been had I specified $ψ_{2} (x)$ , instead of $ψ_{1} (x)$ ?Check that Ehrenfest's theorem holds for this wave function.

d) If you measured the energy of this particle, what values might you get, and with what probabilities?

A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states:

$Ψ (x, 0) = A [ψ_{1} (x) + ψ_{2} (x)]$

You can look up the series

$\frac{1}{1^{6}} + \frac{1}{3^{6}} + \frac{1}{5^{6}} + \dots = \frac{π^{6}}{960}$

and

$\frac{1}{1^{4}} + \frac{1}{3^{4}} + \frac{1}{5^{4}} + \dots = \frac{π^{4}}{96}$

in math tables. under "Sums of Reciprocal Powers" or "Riemann Zeta Function."

(a) Normalize $Ψ (x, 0)$ . (That is, find A. This is very easy, if you exploit the orthonormality of $ψ_{1}$ and $ψ_{2}$ Recall that, having $ψ_{}$ normalized at , $t = 0$ , you can rest assured that is stays normalized—if you doubt this, check it explicitly after doing part(b).

(b) Find $Ψ (x, t)$ and ${| Ψ (x, t) |}^{2}$ Express the latter as a sinusoidal function of time. To simplify the result, let $ω \equiv \frac{π^{2} ℏ}{2 {ma}^{2}}$

c)Compute $⟨ x ⟩$ . Notice that it oscillates in time. What is the angular frequency of the oscillation? What is the amplitude of the oscillation?(If your amplitude is greater than $\frac{a}{2}$ , go directly to jail.

(d) Compute $⟨ p ⟩$

(e) If you measured the energy of this particle, what values might you get, and what is the probability of getting each of them? Find the expectation value of $H$ .How does it compare with E₁ and E₂

A particle of mass m and kinetic energy E > 0 approaches an abrupt potential drop V₀ (Figure 2.19).

(a)What is the probability that it will “reflect” back, if E = V₀/3? Hint: This is just like problem 2.34, except that the step now goes down, instead of up.

(b) I drew the figure so as to make you think of a car approaching a cliff, but obviously the probability of “bouncing back” from the edge of a cliff is far smaller than what you got in (a)—unless you’re Bugs Bunny. Explain why this potential does not correctly represent a cliff. Hint: In Figure 2.20 the potential energy of the car drops discontinuously to −V0, as it passes x = 0; would this be true for a falling car?

(c) When a free neutron enters a nucleus, it experiences a sudden drop in potential energy, from V = 0 outside to around −12 MeV (million electron volts) inside. Suppose a neutron, emitted with kinetic energy 4 MeV by a fission event, strikes such a nucleus. What is the probability it will be absorbed, thereby initiating another fission? Hint: You calculated the probability of reflection in part (a); use T = 1 − R to get the probability of transmission through the surface.

A free particle has the initial wave function
$ψ (x, 0) = {Ae}^{- a | x |},$

where A and a are positive real constants.

(a)Normalize $ψ (x, 0) .$

(b) Find $ϕ (k)$ .

(c) Construct $ψ (x, t)$ ,in the form of an integral.

(d) Discuss the limiting cases very large, and a very small.

See all solutions

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