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    QuantumMechanics Quetions

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    vijay
    This document contains four quantum mechanics questions covering various topics: 1) A particle in a one-dimensional potential well. The energy levels and wavefunctions are derived. 2) The quantum harmonic oscillator. The ground state energy is found to be quantized. 3) A particle confined to a ring. The energy levels are shown to be quantized due to the boundary condition. 4) Wave-particle duality and de Broglie's relation are discussed. Momentum and wavelength calculations are done for photons, electrons, and helium atoms.

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    QuantumMechanics Quetions

    Uploaded by

    vijay
    179 views2 pages
    This document contains four quantum mechanics questions covering various topics: 1) A particle in a one-dimensional potential well. The energy levels and wavefunctions are derived. 2) The quantum harmonic oscillator. The ground state energy is found to be quantized. 3) A particle confined to a ring. The energy levels are shown to be quantized due to the boundary condition. 4) Wave-particle duality and de Broglie's relation are discussed. Momentum and wavelength calculations are done for photons, electrons, and helium atoms.

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    atomic structure

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    This document contains four quantum mechanics questions covering various topics: 1) A particle in a one-dimensional potential well. The energy levels and wavefunctions are derived. 2) The quantum harmonic oscillator. The ground state energy is found to be quantized. 3) A particle confined to a ring. The energy levels are shown to be quantized due to the boundary condition. 4) Wave-particle duality and de Broglie's relation are discussed. Momentum and wavelength calculations are done for photons, electrons, and helium atoms.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    179 views2 pages

    QuantumMechanics Quetions

    Uploaded by

    vijay
    This document contains four quantum mechanics questions covering various topics: 1) A particle in a one-dimensional potential well. The energy levels and wavefunctions are derived. 2) The quantum harmonic oscillator. The ground state energy is found to be quantized. 3) A particle confined to a ring. The energy levels are shown to be quantized due to the boundary condition. 4) Wave-particle duality and de Broglie's relation are discussed. Momentum and wavelength calculations are done for photons, electrons, and helium atoms.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    QUANTUM MECHANICS I

    QUESTION 1
    A particle of mass m is constrained to move in a one-dimensional potential well with V(x)=0 for 0
    infinite elsewhere.

    x L and V(x)

    a) Show that the function


    (x) = a cos(nx/L) + b sin(nx/L),
    where a and b are constants and n is a number, is a solution of the Schrodinger equation for the system.
    b) Using the boundary conditions at x=0 and L, show that a=0 and find the allowed values of n and the
    corresponding energies.
    c) Evaluate the constant b that ensures correct normalisation of the wavefunction (x).
    d) Sketch (x) and (x)2 for the lowest three values of n. Comment on the physical significance of (x) and (x)2.
    e) An electron is confined to a one-dimensional box of length 15 nm. How many energy levels lie between 3.5 and
    8.0 kJ mol-1? (me = 9.110x10-31 kg.)

    QUESTION 2
    The Schrodinger equation for a one-dimensional harmonic oscillator may be written
    2

    d2 1 2
    + kx = E
    2 dx2 2
    where is the effective mass of the oscillator and x is the displacement from the equilibrium position.
    a) Show that the wavefunction = Ae-x2, where A is a constant and = (k)1/2/2 satisfies the Schrodinger
    equation.
    b) Find the energy associated with the wavefunction given in a) and comment on the result obtained.
    c) Make a sketch showing the variation of the potential energy with x. On your diagram, show the energies and
    wavefunctions associated with the four lowest states of the harmonic oscillator. In what ways does the behaviour
    of the quantum harmonic oscillator differ from that expected from classical mechanics?

    QUESTION 3
    The Hamiltonian for a particle of mass m, confined to a ring of radius r, can be written
    H=-

    2 2
    2mr2 2

    where is the angle in the plane of the ring and the potential is a constant, taken as zero.

    a) Show that the wavefunction () = N ein satisfy the Schrodinger equation, and normalise the wavefunctions.
    b) The appropriate boundary conditions for motion on a ring can be written () = (+2). Show that imposition
    of this boundary condition to the wavefunctions given in a) leads to quantised energy levels and derive an
    expression for the energy levels.
    c) What is the Born interpretation of the wavefunction? What conclusions can be drawn by adopting the Born
    interpretation of the wavefunction given in a)?

    QUESTION 4
    a) What is meant by wave-particle duality? State de Broglies relation between wavelength and momentum and
    explain qualitatively the connection between de Broglies relation and the Heisenberg Uncertainty Principle.
    b) Calculate the momentum of a photon of yellow light from a sodium lamp ( = 589 nm)
    c) In low-energy electron diffraction, a beam of electrons is accelerated through a potential of 100 V before
    striking a sample. Show that the momentum, p, of the electron is given by p = 2meeV , where V is the
    accelerating potential, me is the mass of the electron and e the electronic charge. Hence calculate the wavelength
    of the electrons at the sample. Compare this value with the typical spacing between atoms in a metal.
    d) Using the Principle of Equipartition, write down an expression for the average kinetic energy of a helium atom.
    Hence estimate a typical wavelength for a helium atom at 298 K (mHe = 4 g mol-1)

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