Chapter 40: Q. 27 (page 1176)

A $2.0 μm$ diameter water droplet is moving with a speed of $1.0 mm / s$ in a $20 μm$ long box.
a. Estimate the particle’s quantum number.
b. Use the correspondence principle to determine whether quantum mechanics is needed to understand the particle’s motion or if it is “safe” to use classical physics.

Short Answer

Expert verified

a. the quantum number is $2.52 \times 10^{8} .$

b. since it is too large we can use the classical approach.

Step by step solution

Part (a) Step 1: Given information

We have given,

Diameter of the drop = $2 μm$

speed of the drop = $1 mm / s$

length of he box = $20 μm$

we have to find the quantum number n?

Simplify

We know the energy of one dimensional box is

$E = \frac{n^{2} h^{2}}{8 {mL}^{2}} n = {(\frac{8 {mL}^{2} E}{h^{2}})}^{\frac{1}{2}} n = {(\frac{8 (ρ \times \frac{4}{3} {πr}^{3} \times L^{2 \times} \frac{1}{2} {mv}^{2}}{h^{2}})}^{1 / 2} n = (\frac{8 \times 10^{3} kg / m^{3} \times (4 π \times 10^{- 6} m) \times 400 \times 10^{- 12} \times 10^{3} \times 4 π \times 10^{- 6} m \times 10^{- 3} m / s}{3 \times 2 \times 3 \times 6.63 \times 10^{- 34} j / s}) n = 2.52 \times 10^{8}$

Part (b) Step 1: Given information

We have to find its classical approach using corresponds principle.

Simplify

According to the correspondence principle that if any system in quantum mechanics describing the behavior of quantum systems and it must yield the same type of results as classical physics in the macroscopic limit.

then we can approach as same.

From the above found the quantum number is too large.

we can very safely use classical mechanics.

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Short Answer

Step by step solution

Part (a) Step 1: Given information

Simplify

Part (b) Step 1: Given information

Simplify

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A 2.0μmdiameter water droplet is moving with a speed of 1.0mm/sin a 20μmlong box. a. Estimate the particle’s quantum number.b. Use the correspondence principle to determine whether quantum mechanics is needed to understand the particle’s motion or if it is “safe” to use classical physics.

Short Answer

Step by step solution

Part (a) Step 1: Given information

Simplify

Part (b) Step 1: Given information

Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Particle Model of Matter

Electricity

Dynamics

Kinematics Physics

Electromagnetism

Rotational Dynamics

Study anywhere. Anytime. Across all devices.