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Exercises (Problems)

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    4.1: The Wave Theory of Light


    What is the wavelength associated with a photon of a light with the energy is \(3.6 \times 10^{-19}\;J\)?


    Calculate the energy of a photon of a light with the frequency is \(6.5 \times 10^{-14}\; s^{-1}\) ?


    Calculate the energy per photon and the number of photons emitted per minute from

    1. 100-W yellow light bulb (\(λ= 550\;nm\))
    2. 1-kW microwave source (\(λ = 1\;cm\))

    4.3: The Photoelectric Effect


    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


    Calculate the wavenumber of the wavelength of the light emitted from the \(n=8\) to \(n=6\) transition.


    4.5: de Broglie's Postulate


    Calculate the wavelength associated with a 42 g baseball with speed of 80 m/s.


    Calculate the de Broglie wavelengths of the following:

    1. a .8g bullet with velocity 340ms-1.
    2. a 10-5g particle with velocity 10-5ms-1.
    3. a 10-8g particle with velocity 10-8ms-1.
    4. an electron moving with velocity 4.8*106ms-1.


    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


    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?


    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.

    1. What is the momentum of a galloping horse? What is its wavelength?
    2. 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?
    3. 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?


    • de Broglie's postulate deals with the wave-like properties of particles.
    • Heisenberg's uncertainty principle deals with uncertainty of simultaneous measurements.


    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


    What are the results of operating on the following functions with following two operators:

    • \( \hat{A} = \dfrac{d}{dx}\) and
    • \( \hat{B} = \dfrac{d^2}{dx^2}\)
    1. \(3e^{-cx^3}\)
    2. \(\cos(4ax^2)\)
    3. \(e^{i^2kx^3}\)

    4.8: Particle in a One-Dimensional Box




    1. 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.
    2. If an electron makes a transition from n = 3 to n = 1, what will be the wavelength of the emitted radiation?


    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


    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.


    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.


    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.


    How many orbitals have l = 2 and n = 3?

    Exercises (Problems) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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