Exercises (Problems)
- Page ID
- 49051
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)4.1: The Wave Theory of Light
Q4.1.1
What is the wavelength associated with a photon of a light with the energy is \(3.6 \times 10^{-19}\;J\)?
Q4.1.2
Calculate the energy of a photon of a light with the frequency is \(6.5 \times 10^{-14}\; s^{-1}\) ?
Q4.1.3
Calculate the energy per photon and the number of photons emitted per minute from
- 100-W yellow light bulb (\(λ= 550\;nm\))
- 1-kW microwave source (\(λ = 1\;cm\))
4.3: The Photoelectric Effect
Q4.3.1
If the wavelength of a x-ray photon in a x-ray photoelectron spectroscopy (XPS) instrument is 1.25 nm. Calculate the velocity of the electrons emitted from molecules in which the following work functions (e.g., binding energies): 25, 125, 425 eV.
4.4: Bohr's Theory of the Hydrogen Emission Spectrum
Q4.4.2
Calculate the wavenumber of the wavelength of the light emitted from the \(n=8\) to \(n=6\) transition.
Q4.4.3
4.5: de Broglie's Postulate
Q4.5.1
Calculate the wavelength associated with a 42 g baseball with speed of 80 m/s.
Q4.5.2
Calculate the de Broglie wavelengths of the following:
- a .8g bullet with velocity 340ms-1.
- a 10-5g particle with velocity 10-5ms-1.
- a 10-8g particle with velocity 10-8ms-1.
- an electron moving with velocity 4.8*106ms-1.
Q4.5.3
What is the de Broglie wavelength of a thermal neutron at 350 K? A thermal neutron has the kinetic energy equal to the average kinetic energy of a thermalized monotonic gas (i.e., described by the Maxwell-Boltzmann distribution).
4.6: The Heisenberg Uncertainty Principle
Q4.6.1
If the uncertainty of measuring the position of an electron is 2.0 Å, what is the uncertainty of simultaneously measuring its velocity? Hint: What formula deals with uncertainty of measurements?
Q4.6.2
A typical mass for a horse is 510 kg, and a typical galloping speed is 22 kilometers per hour. Use these values to answer the following questions.
- What is the momentum of a galloping horse? What is its wavelength?
- If a galloping horse's velocity and position are simultaneously measured, and the velocity is measured to within ± 1.0%, what is the uncertainty of its position?
- Suppose Planck's constant was actually 0.01 J s. How would that change your answers to (a) and (b)? Which values would be unchanged?
Hints:
- de Broglie's postulate deals with the wave-like properties of particles.
- Heisenberg's uncertainty principle deals with uncertainty of simultaneous measurements.
Q4.6.3
Consider a balloon with a diameter of \(2.5 \times 10^{-5}\; m\). What is the uncertainty of the velocity of an oxygen molecule that is trapped inside.
4.7: The Schrödinger Wave Equation
Q4.7.1
What are the results of operating on the following functions with following two operators:
- \( \hat{A} = \dfrac{d}{dx}\) and
- \( \hat{B} = \dfrac{d^2}{dx^2}\)
- \(3e^{-cx^3}\)
- \(\cos(4ax^2)\)
- \(e^{i^2kx^3}\)
4.8: Particle in a One-Dimensional Box
Q4.8.1
Q4.8.2
Q4.8.3
- Calculate the energy levels for n = 1, 3, and 5 for an electron in a potential well of width 0.50 nm with infinite barriers on either side.
- If an electron makes a transition from n = 3 to n = 1, what will be the wavelength of the emitted radiation?
Q4.8.4
For a helium atom in a one-dimensional box, calculate the quantum number for the wavefunctions with the energies equal to the average kinetic energy of a thermalized monotonic gas (e.g, 3/2 kT) for a box 1 nm long at -100 oC, 0 oC, and 100 oC.
4.10: The Schrödinger Wave Equation for the Hydrogen Atom
Q4.10.1
Indicate the number of subshells, the number of orbitals in each subshell, and the values of l and ml for the orbitals in the n = 4 shell of an atom.
Q4.10.2
Identify the subshell in which electrons with the following quantum numbers are found: (a) n = 3, l = 1; (b) n = 5, l = 3; (c) n = 2, l = 0.
Q4.10.3
Calculate the maximum number of electrons that can occupy a shell with (a) n = 2, (b) n = 5, and (c) n as a variable. Note you are only looking at the orbitals with the specified n value, not those at lower energies.
Q4.10.4
How many orbitals have l = 2 and n = 3?