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    The Morse Potential

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    Pooja Sharma
    The Morse potential is used to model vibrational excitations of a chemical bond. It defines the Hamiltonian of the system in terms of the bond length, conjugate momentum, bond strength, and variation about equilibrium. By making substitutions and analogizing to the 2D harmonic oscillator, the eigenvalues of the Hamiltonian are found to correspond to actual bound states of the Morse potential up to a point, with higher eigenvalues representing unbound states.

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    The Morse Potential

    Uploaded by

    Pooja Sharma
    199 views1 page
    The Morse potential is used to model vibrational excitations of a chemical bond. It defines the Hamiltonian of the system in terms of the bond length, conjugate momentum, bond strength, and variation about equilibrium. By making substitutions and analogizing to the 2D harmonic oscillator, the eigenvalues of the Hamiltonian are found to correspond to actual bound states of the Morse potential up to a point, with higher eigenvalues representing unbound states.

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    The Morse potential is used to model vibrational excitations of a chemical bond. It defines the Hamiltonian of the system in terms of the bond length, conjugate momentum, bond strength, and variation about equilibrium. By making substitutions and analogizing to the 2D harmonic oscillator, the eigenvalues of the Hamiltonian are found to correspond to actual bound states of the Morse potential up to a point, with higher eigenvalues representing unbound states.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    199 views1 page

    The Morse Potential

    Uploaded by

    Pooja Sharma
    The Morse potential is used to model vibrational excitations of a chemical bond. It defines the Hamiltonian of the system in terms of the bond length, conjugate momentum, bond strength, and variation about equilibrium. By making substitutions and analogizing to the 2D harmonic oscillator, the eigenvalues of the Hamiltonian are found to correspond to actual bound states of the Morse potential up to a point, with higher eigenvalues representing unbound states.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
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    quantum mechanics → harmonic oscillator FFF

    The Morse Potential


    The Morse potential is used to model the vibrational excitations of a chemical bond. If the length of the
    chemical bond is x, with conjugate momentum p, then the Hamiltonian of the system is given by

    p2  
    H= + D e−2x/a − 2e−x/a .
    2µ
    µ roughly corresponds to the “mass” of the two atoms in the bond in the center of mass frame, D is a
    measure of the strength of the chemical bond, and a is a measure of the possibility of variations about
    equilibrium.

    (a) Plot or sketch the Morse potential (the potential energy part of the Hamiltonian) as a function of x.
    Verify that the statements made about D and a are reasonable.

    It is tricky to find the eigenvalues E of a Hamiltonian like this, in general. However, here we can do
    it with a trick. Begin by defining the dimensionless variables

    2µEa2
    ≡ ,
    ~2
    2µDa2
    κ2 ≡ ,
    ~2
    √ −x/2a
    r ≡ κe .

    (b) Write Schrödinger’s equation in position space, make the substitutions above, and show that it be-
    comes
    d2 ψ 1 dψ r2 4
    2
    + + ψ + 2 ψ = 2κψ.
    dr r dr 2 r
    (c) Make an analogy to the 2D harmonic oscillator, and conclude that the eigenvalues En of H (now in
    dimensionful units) are given by
      2
    ~ 1
    En = −D 1 − √ n+ .
    a 2µD 2

    (d) Show that at some point, En ≥ En+1 . Argue that only the eigenvalues E0 , E1 , . . . En correspond to
    actual bound states for H.

    (e) There is a critical value of D, Dc , such that if D ≤ Dc , there are no bound states to the Morse
    potential. Find the value of Dc .

    (f) On the other hand, if D  Dc , then we can approximate that the first few excited states will
    approximately look like a harmonic oscillator spectrum (in 1D). Express the effective frequency of
    this oscillator, ωeff , in terms of ~, a, D and µ. What is the physical reason why this approximation
    is valid?

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