Q: A metal with a work function $$\phi = 2.5\ \text{eV} is $$illuminated with light of frequency $$\nu = 1.0 \times 10$$^{15}$$\ \text{Hz}. $$Calculate the maximum kinetic energy of the emitted electrons. ($$h = 6.626 \times 10$$^{-34}$$\ \text{J s}$$, $$1\ \text{eV} = 1.602 \times 10$$^{-19}$$\ \text{J})$$

$$K_{\text{max}} = 2.6\ \text{eV}$$

Q: For an electron in a hydrogen atom with velocity $$v = 2.2 \times 10^6$\ \text{m/s}$$ and mass $$m = 9.109 \times 10$$^{-31}$$\ \text{kg}$$, what is its de Broglie wavelength?

$$\lambda \approx 0.33\ \text{nm}$$

Question 1

Which phenomenon led to the introduction of the concept of quantized energy levels in physics, marking the birth of quantum mechanics?

Accepted Answer

Blackbody radiation

Question 2

According to Planck’s quantum hypothesis, what is the energy of a quantum of radiation?

Accepted Answer

$$E = h
u$$

Question 3

Which classical law predicted the 'ultraviolet catastrophe' in blackbody radiation?

Accepted Answer

Rayleigh-Jeans Law

Question 4

A metal with a work function $$\phi = 2.5\ 	ext{eV} is $$illuminated with light of frequency $$
u = 1.0 	imes 10$$^{15}$$\ 	ext{Hz}. $$Calculate the maximum kinetic energy of the emitted electrons. ($$h = 6.626 	imes 10$$^{-34}$$\ 	ext{J s}$$, $$1\ 	ext{eV} = 1.602 	imes 10$$^{-19}$$\ 	ext{J})$$

Accepted Answer

$$K_{	ext{max}} = 2.6\ 	ext{eV}$$

Question 5

Which observation from the photoelectric effect experiment contradicts classical physics predictions?

Accepted Answer

Electrons are emitted instantaneously when light of sufficient frequency strikes the metal

Question 6

The de Broglie wavelength of a particle is given by:

Accepted Answer

$$\lambda = \frac{h}{p}$$

Question 7

Which principle states that it is impossible to simultaneously know the exact position and momentum of a particle?

Accepted Answer

Heisenberg’s Uncertainty Principle

Question 8

For an electron in a hydrogen atom with velocity $$v = 2.2 	imes 10^6$\ 	ext{m/s}$$ and mass $$m = 9.109 	imes 10$$^{-31}$$\ 	ext{kg}$$, what is its de Broglie wavelength?

Accepted Answer

$$\lambda \approx 0.33\ 	ext{nm}$$

Question 9

Why is the de Broglie wavelength of a baseball (mass $$0.15\ 	ext{kg}$$, velocity $$40\ 	ext{m/s}) $$negligible compared to that of an electron?

Accepted Answer

Because the baseball’s mass is much larger, making its wavelength extremely small

Question 10

In the double-slit experiment, what happens to the interference pattern if you observe which slit the particle passes through?

Accepted Answer

The interference pattern disappears

Question 11

The wave function $$\psi(x, t)$$ must satisfy which condition for normalization?

Accepted Answer

$$\int_{-\infty}^{\infty} |\psi(x, t)|^2 dx = 1$$

Question 12

Born’s interpretation of the wave function states that $$|\psi(x, t)|^2$$ represents:

Accepted Answer

The probability density of finding the particle at position $$x$$ and time $$t$$

Question 13

Which equation describes the conservation of probability in quantum mechanics?

Accepted Answer

$$\frac{\partial \rho(x, t)}{\partial t} + \frac{\partial \vec{j}(x, t)}{\partial x} = 0$$

Question 14

For a free particle described by $$\psi(x, t) = A e$$^{i(kx - \omega t)}$$, what is the probability current density?

Accepted Answer

$$\vec{j}(x, t) = \frac{\hbar k}{m} |A|^2$$

Question 15

Which property is NOT required for a physically acceptable wave function?

Accepted Answer

It must be infinite at some points

Question 16

The probability current density $$\vec{j}(x, t) in $$quantum mechanics is analogous to which quantity in electromagnetism?

Accepted Answer

Current density

Question 17

Which of the following best describes the complementarity principle?

Accepted Answer

Particles and waves are complementary aspects of the same entity, depending on the experimental setup

Question 18

If a wave function is given by $$\psi(x) = A e$$^{-\alpha |x|}$$, what is the normalization constant A$?

Accepted Answer

$$A = \sqrt{\alpha}$$

Question 19

Which equation is the time-dependent Schrödinger equation for a particle in one dimension?

Accepted Answer

$$i\hbar \frac{\partial \psi}{\partial t} = -\frac{\$$hbar^2$$}{2m} \frac{\$$partial^2$$ \psi}{\partial x^2$} + V(x)\psi$$

Quantum Mechanics Foundations: Blackbody Radiation, Photoelectric Effect, and Wave-Particle Duality

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