An Harmonic

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Anharmonicity In real molecules, highly sensitive vibrational spectroscopy can detect overtones , which are transitions originating from the n = 0 state for which Δn = +2, +3, … Overtones are due to anharmonicity. A good approximation of realistic anharmonicity is given by the Morse potential. V ( r)= D e 1− e α ( r r 0 ) [ ] 2

description

Harmonic oscillators, quantum mechanics

Transcript of An Harmonic

Page 1: An Harmonic

Anharmonicity

In real molecules, highly sensitive vibrational spectroscopy can detect overtones, which are transitions originating from the n = 0 state for which Δn = +2, +3, …

Overtones are due to anharmonicity. A good approximation of realistic anharmonicity is given by the Morse potential.

V (r) = De 1− e−α (r−r0 )[ ]

2

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So we do

Put x = r – r0 and Taylor expand:

V (r) = De 1− e−α (r−r0 )[ ]

2

V (x) = De 1− 2e−αx + e−2αx( )

=Deα2x 2

Comparing to the harmonic oscillator

V (x) = 12 kx

2

we see that

k = 2Deα2

≈De 1− 2 1−αx + 12α

2x 2[ ] + 1− 2αx + 1

2 4α 2x 2[ ]( )

De → cDe

α →αc

to keep the force constant the same but change the anharmonicity€

α =k

2De

⎝ ⎜

⎠ ⎟

1/ 2

=μω2

2De

⎝ ⎜

⎠ ⎟

1/ 2

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V (x) = De 1− e−αx[ ]

2 use De = 40, α = 1; then scale by c

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Energy levels

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Morse model

En = 2αDe1/ 2 n +1/2( ) − n +1/2( )

2 α

2De1/ 2

⎣ ⎢

⎦ ⎥

ψn (x) = Nnyβ −n−1/ 2e−y / 2Ln

2β −2n−1(y)

β =De

1/ 2

α

y = 2βe−αx

Nn =α (2β − 2n −1)n!

Γ(2β − n)

⎣ ⎢

⎦ ⎥

1/ 2

Lna (x) are the generalized Laguerre polynomials

0 ≤ n <De

1/ 2

α−

1

2dissociated above this

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Harmonic oscillator model

ψn (x) = NnHn (α 1/ 2x)e−αx 2 / 2

α =kμ

h2

⎝ ⎜

⎠ ⎟

1/ 2

Nn =1

2n n!( )1/ 2

α

π

⎝ ⎜

⎠ ⎟

1/ 4

are the Hermite polynomials

Hn (x)

EnMORSE = 2αDe

1/ 2 n +1/2( ) − n +1/2( )2 α

2De1/ 2

⎣ ⎢

⎦ ⎥

EnHO = 2αDe

1/ 2 n +1/2( )

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Wavefunctions: harmonic oscillator

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Wavefunctions: Morse oscillator

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Wavefunctions: harmonic vs. Morse

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Wavefunctions

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Wavefunctions

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Expectation value of position

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Expectation value of position

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Expectation value of position

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Selection rules

μz( )nn'= ψ n'

*μ zψ n−∞

μz ≈ μ0 +dμ

dx 0

x +d2μ

dx 2

0

x 2

2!+L

For anharmonicity, can replace the H.O. wavefunctions with Morse wavefunctions…

…or can keep more terms in the Taylor expansion of the dipole moment

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Selection rules

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Pclassical (x) = π 2E /k − x 2( )

−1

2E /k = x turn2

Where xturn is the maximum value of x

Correspondence principle

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Correspondence principle

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Correspondence principle

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Correspondence principle