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

$唯 (x, 0) = A [蠄_{1} (x) + 蠄_{2} (x)]$

You can look up the series

$\frac{1}{1^{6}} + \frac{1}{3^{6}} + \frac{1}{5^{6}} + 鈰� = \frac{蟺^{6}}{960}$

and

$\frac{1}{1^{4}} + \frac{1}{3^{4}} + \frac{1}{5^{4}} + 鈰� = \frac{蟺^{4}}{96}$

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

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

(b) Find $唯 (x, t)$ and ${| 唯 (x, t) |}^{2}$ Express the latter as a sinusoidal function of time. To simplify the result, let $蝇鈮� \frac{蟺^{2} 鈩�}{2 {ma}^{2}}$

c)Compute $鉄� x 鉄�$ . 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 $鉄� p 鉄�$

(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) 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?

The gaussian wave packet. A free particle has the initial wave function

$Y (x, 0) = A e^{- a x^{2}}$

where $A$ and are constants ( is real and positive).

(a) Normalize $Y (x, 0)$

(b) Find $Y (x, t)$ . Hint: Integrals of the form

${鈭�}_{- 鈭�}^{+ 鈭�} e^{- (a x^{2} + b x)} d x$

Can be handled by 鈥渃ompleting the square鈥�: Let $y = \sqrt{a} [x + (b l 2 a)]$ , and note that $(a x^{2} + b x) = y^{2} - (b^{2} l 4 a)$ . Answer:

localid="1658297483210" $Y (x, t) = {(\frac{2 a}{蟺})}^{1 / 4} \frac{e^{- e x^{2} l [1 + (2 i h a t l m)]}}{\sqrt{1 + (2 i h a t l m)}}$

(c) Find . Express your answer in terms of the quantity

localid="1658297497509" $蝇 = \sqrt{\frac{a}{1 + {(2 i h at l m)}^{2}}}$

Sketchlocalid="1658124147567" ${| Y |}^{2}$ (as a function of $x$ ) at $t = 0$ , and again for some very large $t$ . Qualitatively, what happens to ${| Y |}^{2}$ , as time goes on?

(d) Find $< x >, < p >, < x^{2} >, < p^{2} >, 蟽_{x}$ and $蟽_{P}$ . Partial answer:localid="1658297458579" $< p^{2} > = a h^{2}$ , but it may take some algebra to reduce it to this simple form.

(e) Does the uncertainty principle hold? At what time $t$ does the system come

closest to the uncertainty limit?

-consider the 鈥渟tep鈥� potential:

$v (x) = {\begin{matrix} 0, & if x 鈮� 0, \\ V_{0}, & if x > 0, \end{matrix}$

a.Calculate the reflection coefficient, for the case E < V₀, and comment on the answer.

b. Calculate the reflection coefficient, for the case E >V₀.

c. For potential such as this, which does not go back to zero to the right of the barrier, the transmission coefficient is not simply $\frac{|F^{2}|}{|A^{2}|}$ (with A the incident amplitude and F the transmitted amplitude), because the transmitted wave travels at a different speed . Show that $T = \sqrt{\frac{E - V_{0}}{V_{0}}} \frac{|F^{2}|}{|A^{2}|}$ ,for E >V₀. What is T for E < V₀?

d. For E > V_0, calculate the transmission coefficient for the step potential, and check that T + R = 1.

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 鈥� the right wall moving from a to 2a 鈥� 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)

See all solutions

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