Quantum Mechanics Answers

Equation of Quantum hormonic oscillator

Quantum mechanics is relevant, when the de Broglie wavelength of the particle is greater than the distance between particles. The purpose of this problem is to determine which systems will have to be treated quantum mechanically and which can be described classically.

a) Show that the typical de Broglie wavelength of a particle in an ideal gas in

equilibrium is 𝜆 =

ℎ

√3𝑚𝐾𝐵𝑇

b) Solids: The lattice spacing in a typical solid is d = 0.3 nm. Find the temperature below which the free electrons in a solid are quantum mechanical? (Hint: Refer the a) part of the question and treat free electrons as a gas and the lattice spacing as the typical distance between the electrons)

c) Gases: For what temperatures are the atoms in an ideal gas at pressure 𝑃 quantum mechanical? (Hint: Use the ideal gas law, to deduce the inter atomic distance)

Is Helium at atmospheric pressure quantum mechanical?

What about Hydrogen atoms in outer space (interatomic distance is 1 cm and temperature is 3 K)?

Quantum mechanics is relevant, when the de Broglie wavelength of the particle is greater than the distance between particles. The purpose of this problem is to determine which systems will have to be treated quantum mechanically and which can be described classically.

a)Gases: For what temperatures are the atoms in an ideal gas at pressure 𝑃 quantum  mechanical? (Hint: Use the ideal gas law, to deduce the inter atomic distance)

Below what temperature, is Helium at atmospheric pressure quantum mechanical?

Below what temperature is Hydrogen atoms in outer space quantum mechanical? 

(interatomic distance is 1 cm and temperature is 3 K)

With a neat diagram, explain the working of electron microscopes. Briefly explain the role of each experimental component. Compare the resolution of electron microscopes and optical microscopes. 

A particle strikes a potential barrier of height U and width L. Derive an expression for 

the approximate transmission probability, if the energy of the particle E < U.

With a neat diagram, explain the working of electron microscopes. Briefly explain the 

 role of each experimental component. Compare the resolution of electron microscopes 

 and optical microscopes.

Describe one method by which the speed of light can be measured, explaining how the result is calculated from the observations. Describe one practical situation in which a knowledge of the speed

       of light, or other electromagnetic radiation is needed.

Electrons are emitted with zero velocity from a certain metal surface when it is exposed to radiations of lamda=6800A. Calculate threshold frequency and work function of metal.

Consider the Mungo River where it is 200

meters wide and has a uniform flow. A small

boat sailing at full speed takes 5 minutes to

cover 100 meters when sailing upstream, but

only 2 minutes when sailing downstream. What

is the shortest time in which the boat can cross

the Mungo to the nearest point on the opposite

bank?

 Show that < 𝑥𝑝 > − < 𝑝𝑥 > = 𝑖ħ for the ground state wave function of the Quantum 

Harmonic Oscillator (QHO)