Quantum Physics Study Questions

PHYS 252Dr. Varriano

1) Sketch the intensity vs. wavelength curves for blackbody radiation obtained from classical wave theory and Planck's quantum theory. Explain why the classical curve is incorrect. What was Planck's key assumption in his quantum model? (What did he quantize?)

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The classical (Rayleigh-Jeans) theory works okay for longer wavelengths but predicts that the intensity should continue to increase as the wavelength decreases.

Planck assumed that an atom oscillating at frequency f can only have certain energies given by E = nhf. He quantized the energy of an atom, and in so doing, also quantized the emitted electromagnetic radiation into bundles of energy (photons).

2) What is the photoelectric effect? Give at least two observed characteristics of the photoelectric effect that cannot be explained by the classical wave theory of light. Describe how the photon model explains these characteristics.

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Light hits material and electron is ejected. (The electron is called a “photoelectron”.)

Observation 1For frequencies below a certain cut-off frequency, no photoelectrons are seen.Explanation: Since one electron interacts with one photon at a time, the one photon must have sufficient energy (equal to the work function of the material) to eject the electron. The energy of a photon is Eph= hf so there is a minimum frequency.

Observation 2The stopping potential, and therefore the maximum kinetic energy of a photoelectron, is independent of light intensity.Explanation: Since one electron interacts with one photon at a time, the energy of one photon determines the maximum kinetic energy according to Kmax = Eph-. Increasing the intensity increases the number of photons but nit the energy of a single photon.

Observation 3A plot of stopping potential versus frequency is linear.Explanation: Since one electron interacts with one photon at a time, the energy of one photon determines the maximum kinetic energy according to the linear equation Kmax = Eph- = hf-. Since Kmax= e Vs, a plot of Vs vs. f is also linear.

Observation 4Photoelectrons are emitted in a very short time even in very low light intensities.Explanation It only takes one photon to eject an electron.

3) What is Compton scattering? Is Compton scattering explained by the classical wave model of light or by the photon model?

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Scattering of x-rays off of electrons.

Explained by the photon model.

4) List Bohr's four postulates in his model of the Hydrogen atom. What physical quantities are quantized?

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1. Circular orbits via the electric (Coulomb) force.

2. Only certain stable orbits exist with distinct atomic energies.

3. Radiation in the form of a photon is emitted when the electron makes a transition from a higher orbit to a lower orbit. Radiation in the form of a photon can be absorbed and the electron makes a transition from a lower orbit to a higher orbit.

4. These stable orbits are determined by a quantized orbital angular momentum (L = mvr) according to

 

 where the quantum number n = 1,2,3,… and ħ = h / (2π) (pronounced “h-bar”)

5) Describe a physical phenomenon that supports the wave nature of light. Describe a physical phenomenon that supports the particle nature of light.

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Wave Nature of Light• interference (Young’s double slit, thin film, interferometers)• diffraction

Particle Nature of Matter (Photons) • thermal radiation• photoelectric effect• light emission and absorption of atoms• Compton scattering

6) Describe a physical phenomenon that supports the particle nature of matter. Describe a physical phenomenon that supports the wave nature of matter.

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Particle Nature of Matter• All of Physics I and II ! Objects behaving as regular objects.

Wave Nature of Matter• diffraction of electrons• quantum atomic models • quantum tunneling

7) Why aren't we aware of the wave nature of macroscopic objects?

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Their matter wavelengths are just too small to detect. = h / (mv)

8) Describe the Heisenberg Uncertainty Principle (HUP) either in terms of a simultaneous measurement of position and momentum or in terms of a simultaneous measurement of energy and time. What is it about particles that gives rise to the HUP?

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Suppose a particle is moving in one dimension and you measure its position and momentum at the same time. The position measurement has an uncertainty x so that the particle’s position is x ± x. The momentum measurement has an uncertainty p so that the particle’s momentum is p ± p. HUP states

  Suppose you measure the energy of a particle and the time that it has that energy. The energy measurement has an uncertainty E so that the particle’s energy is E ± E. The time measurement has an uncertainty t so that the time is t ± t. HUP states

It is because a particle acts as a wave that gives rise to these two principles.

9) Write the one-dimensional time-independent Schroedinger equation. From where does the equation come?

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where U(x) is the potential energy of the particle and E is the total energy of the particle. There are only certain values of E and certain eigenfunctions (x) that are solutions to this differential equation. The particle’s energies are “quantized”.

The equation is derived from a combination of the law of conservation of energy and the classical wave equation.

10) Write down an integral expression for the probability of finding a particle traveling in one dimension in the region between x=a and x=b.

Write down an integral expression for the normalization condition of a one-dimensional wave function.

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Probability of finding the particle between a and b at time t =

Normalization Condition:

11) List the 4 quantum numbers of an atom. Describe what each quantizes and give the range of possible values.

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Symbol Name What it quantizes

How it quantizes Allowed values

n principle energy E where Zeff depends on n and l

n = 1, 2, 3,…

l orbital orbital angular momentum L

l = 0, 1,…, n-1

ml orbital magnetic

z-component of orbital angular momentum Lz

ml = -l, -l+1, .., 0, .., l

s spin spin angular momentum S

s = 1/2

ms spin magnetic

z-component of spin angular momentum Sz

ms = -1/2 or +1/2

Quantum mechanics applied to atoms give rise to the following five quantum numbers. Since the spin quantum number (s) is always ½, you only need the remaining four quantum numbers to describe an atomic state.

A quantum state is given by a set of quantum numbers of the form (n, l, ml, ms). For a given n, there are 2n2 atomic states.

12) State the Pauli Exclusion Principle. Describe how it helps explain the periodic table of elements.

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“No two electrons in an atom can have the same set of quantum numbers.” or “No two electrons in an atom can be in the same state.”

The exclusion principle helps explain the way that orbitals, subshells, and shells are filled with electrons in atoms. This is a major factor in explaining the organization and structure of the periodic table.

13) Sketch a typical intensity vs. wavelength spectrum obtained in the production of x-rays. What determines the minimum wavelength? What is the source of the characteristic x-ray lines?

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The minimum wavelength is solely determined by the accelerating voltage.

Electrons are ejected from the K-shell (n=1) of atoms in the collisions. The K characteristic lines are photons emitted from electron transitions from higher shells to the vacant K-shells. Kα is a transition from the L-shell (n = 2→1). Kβ is a transition from the M-shell (n = 3→1).

Ka

Kb

14) What are the 3 necessary elements to construct a laser? What does the acronym "laser" stand for?

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1. Gain medium – material with atoms that can undergo stimulated emission2. Excitation Source – gives energy to atom to raise its energy level3. Mirrors to form a resonating cavity – allows some photons to remain in cavity so to keep stimulating atomic transitions

Light Amplification by Stimulated Emission of Radiation

15) Explain how a 3 level laser works. Make sure you know what is meant by a metastable state, population inversion, spontaneous emission, and stimulated emission.

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1

2

31.Pump atom from 1 to 3 using excitation source.

2. Quick decay from 3 to 2.

3. Atom “waits” in metastable state for incoming photon*.

5. Repeat Step 1.

4. Incoming photon stimulates emission from 2 to 1.

metastable state – state where stimulated emission lifetime is longer than spontaneous emission lifetime, state where stimulated emission is more likely than spontaneous emission

population inversion – when there are more atoms in state 2 than in state 1, a necessary condition for continued lasing

spontaneous emission– the emission of a photon from the atom due to a transition from a higher to a lower energy level without the aid of another photon

stimulated emission– the emission of a photon from the atom due to a transition from a higher to a lower energy level that is caused by the interaction of the atom with an incident photon of the same energy as the emitted photon