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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition · 469 pages

ISBN: 9780131118928

Chapter 2: Q30P (page 83)

Normalize $ψ (x)$ the equation 2.151, to determine the constants D and F.

Short Answer

Expert verified

$D = \frac{1}{\sqrt{a + \frac{1}{k}}}$

$F = \frac{e^{ka} cosla}{\sqrt{a + \frac{1}{k}}}$

Step by step solution

01

Define the Schrödinger equation

A differential equation 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 Constants

$= 2 \int_{0}^{\infty} | ψ |^{2} d x = 2 (| D |^{2} \int_{0}^{a} \cos^{2} l x + | F |^{2} \int_{0}^{\infty} e^{- 2 k x} d x)$

$= 2 [{| D |^{2} (\frac{x}{2} + \frac{1}{4 l} \sin 2 l x) |}_{0}^{a} + {{| | F |}^{2} (- \frac{1}{2 k} e^{- 2 k x}) |}_{a}^{\infty} 2 [| D |^{2} (\frac{a}{2} + \frac{\sin 2 l a}{4 l}) + | F |^{2} \frac{e^{- 2 k a}}{2 k}]$

But $F = {De}^{ka} cosla (Eq \cdot 2 .152)$ , so role="math" localid="1656054797733" $1 = | D |^{2} (a + \frac{\sin (2 l a)}{2 l} + \frac{\cos^{2} (l a)}{k})$

Furthermore $k = l \tan (l a) (Eq .2154)$ , so

$= | D |^{2} (a + \frac{2 \sin l a \cos l a}{2 l} + \frac{\cos^{3} l a}{l \sin l a})$

$= | D |^{2} [a + \frac{\cos l a}{l \sin l a} (\sin^{2} l a + \cos^{2} l a)]$

$= | D |^{2} (a + \frac{1}{l \tan l a})$

$= | D |^{2} (a + \frac{1}{k})$

$D = \frac{1}{\sqrt{a + \frac{1}{k}}}$

$F = \frac{e^{k a} \cos l a}{\sqrt{a + \frac{1}{k}}}$

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Most popular questions from this chapter

Consider the moving delta-function well: $V (x, t) = - αδ (x - vt)$

where v is the (constant) velocity of the well. (a) Show that the time-dependent Schrödinger equation admits the exact solution $ψ (x, t) = \sqrt{\frac{mα}{h}} e^{- mα | x - vt |} l h^{2} e^{- i} [(E + (1 / 2) {mv}^{2}) t - mvx] l h$ where $E = - {mα}^{2} l 2 h^{2}$ is the bound-state energy of the stationary delta function. Hint: Plug it in and check it! Use the result of Problem 2.24(b). (b) Find the expectation value of the Hamiltonian in this state, and comment on the result.

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

A particle of mass m is in the ground state of the infinite square well (Evaluation 2.19). Suddenly the well expands to twice its original size – the right wall moving from a to 2a – leaving the wave function (momentarily) undisturbed. The energy of the particle is now measured.

What is the most probable result? What is the probability of getting that result?
What is the next most probable result, and what is its probability?
What is the expectation value of the energy? (Hint: if you find yourself confronted with an infinite series, try another method)

Show that there is no acceptable solution to the Schrodinger equation for the infinite square well with $E = 0$ or $E < 0$ (This is a special case of the general theorem in Problem 2.2, but this time do it by explicitly solving the Schrodinger equation, and showing that you cannot meet the boundary conditions.)

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

See all solutions

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