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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 2: Q18P (page 66)

Show that $[A e^{i k x} + B e^{- i k x}]$ and $[C c o s (k x) + D s i n (k x)]$ are equivalent ways of writing the same function of $x$ , and determine the constants $C$ and $D$ in terms of $A$ and $B$ , and vice versa.

Short Answer

Expert verified

The constants $C$ and $D$ in terms of $A$ and $B$ , and vice versa are,

(i) $C = A + B$ and $D = i (A - B)$

(ii) $A = \frac{1}{2} (C - i D); B = \frac{1}{2} (C + i D)$

Step by step solution

01

Step 1:Formulae used

The required formula are

$e^{ikx} = (c o s (k x) + i s i n (k x))$

$e^{- i k x} = (c o s (k x) - i s i n (k x))$

$c o s (k x) = (\frac{e^{i k x} + e^{- i k x}}{2})$

$s i n (k x) = D (\frac{e^{i k x} - e^{- i k x}}{2})$

02

Compute C and D in terms of A and B and vice versa

Apply Euler's formula,

$A e^{i k x} + B e^{- i k x} = A (\cos (k x) + i \sin (k x)) + B (\cos (k x) - i \sin (k x))$

$= (A + B) \cos (k x) + i (A - B) \sin (k x)$

$= C \cos (k x) + D \sin (k x)$

Thus, $C$ and $D$ expressed in terms of arbitrary constants $A$ and $B$ , $C = A + B$

$D = i (A - B)$

Now, Solve both the equations for $A$ and B and get,

$C \cos (k x) + D \sin (k x) = C (\frac{e^{i k x} + e^{i k x}}{2}) + D (\frac{e^{i k x} - e^{k x}}{2 i})$

$= \frac{1}{2} (C - i D) e^{i k x} + \frac{1}{2} (C + i D) e^{i k x}$

$= A e^{A x} + B e^{i x x}$

Therefore, the values of $A$ and $B$ are,

A = \frac{1}{2} (C - i D)

B = \frac{1}{2} (C + i D)

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Most popular questions from this chapter

A particle is in the ground state of the harmonic oscillator with classical frequency $蝇$ , when suddenly the spring constant quadruples, so $蝇' = 2 蝇$ , without initially changing the wave function (of course, $唯$ will now evolve differently, because the Hamiltonian has changed). What is the probability that a measurement of the energy would still return the value $\frac{鈩� 蝇}{2}$ ? What is the probability of getting $鈩� 蝇$ ?

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

a) Find $A$ .

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

c) Find $鉄� x 鉄�$ and $鉄� p 鉄�$ . Don't get too excited if they oscillate at the classical frequency; what would it have been had I specified $蠄_{2} (x)$ , instead of $蠄_{1} (x)$ ?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) Show that the wave function of a particle in the infinite square well returns to its original form after a quantum revival time T = 4ma2/蟺~. That is: 唯 (x, T) = 唯 (x, 0) for any state (not just a stationary state).

(b) What is the classical revival time, for a particle of energy E bouncing back and forth between the walls?

(c) For what energy are the two revival times equal?

A particle of mass m in the harmonic oscillator potential (Equation 2.44) starts $蠄 (x, 0) = A {(1 - 2 \sqrt{\frac{m 蝇}{魔}} x)}^{2} e^{- \frac{m 蝇}{2 魔} x^{2}}$ 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" $蠄 (x, T) = B {(1 + 2 \sqrt{\frac{m 蝇}{魔}} x)}^{2} e^{- \frac{m 蝇}{2 魔} x^{2}}$
for some constant B. What is the smallest possible value of T ?

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

See all solutions

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