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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 2: Q19P (page 66)

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

Short Answer

Expert verified

The probability current $(J)$ is

j = \frac{\hat{鈩�} k}{m} | A |^{2}

Step by step solution

01

The probability current density (J)

The required formula of probability current density $(J)$ is,

role="math" localid="1657786971189"

J = \frac{i 鈭�}{2 m} (唯 \frac{鈭� 唯^{'}}{鈭� x} - 唯^{*} \frac{鈭� 唯}{鈭� x})

Where $y$ is the wave function of the particle and complex conjugate of $y^{is} y 掳$ .

02

Compute probability current density

Equation $2.94$ from the chapter is

$唯_{k} (X, t) = A e^{i k x^{j k k^{2}}} 2 m^{2}$

蠄^{'} = A e^{i k \frac{k m^{2}}{2 m^{2}} ?},

Thus, it can written that,

$J = \frac{i 鈫�}{2 m} (唯 \frac{鈭� 唯^{*}}{鈭� x} - 唯^{*} \frac{鈭� 唯}{鈭� x})$

$= \frac{i 脳}{2 m} | A |^{2} [e^{i (k x - \frac{x k^{2}}{2 m} t} (- i k) e^{- i (k x - \frac{a k^{2}}{2 m} t} - e^{(\{k x \frac{x k^{2} t}{2 m} t}) (i k) e^{(k x - \frac{m k^{2}}{2 m} t)}]$

$J = \frac{i 鈫�}{2 m} | A |^{2} (- 2 i k)$

$J = \frac{鈮� k}{m} | A |^{2}$

It flows in the positive $(x)$ direction.

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Most popular questions from this chapter

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

where A and a are positive real constants.

(a)Normalize $蠄 (x, 0) .$

(b) Find $蠒 (k)$ .

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

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

Solve the time-independent Schr 虉odinger equation for a centered infinite square well with a delta-function barrier in the middle:

$V (x) = {\begin{cases} 伪未 (x) for - a < x < + a \\ 鈭� for | x | 鈮� a \end{cases}$

Treat the even and odd wave functions separately. Don鈥檛 bother to normalize them. Find the allowed energies (graphically, if necessary). How do they compare with the corresponding energies in the absence of the delta function? Explain why the odd solutions are not affected by the delta function. Comment on the limiting cases 伪鈫� 0 and 伪鈫� 鈭�.

Although the overall phase constant of the wave function is of no physical significance (it cancels out whenever you calculate a measurable quantity), the relative phase of the coefficients in Equation 2.17 does matter. For example, suppose we change the relative phase of $蠄_{1} and 蠄_{2}$ in problem 2.5: $蠄 (x, 0) = A [蠄_{1} (x) + e^{i 蠒} 蠄_{2} (x)]$ Where $蠒$ is some constant. Find $蠄 (x, t), {| 蠄 (x, t) |}^{2}$ , and $(x)$ , and compare your results with what you got before. Study the special cases $蠒 = \frac{蟺}{2} and 蠒 = 蟺$ .

-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.

See all solutions

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