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Momentum representation, momentum space wave function

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We study the relationship between the position space wave function and the momentum space wave function in quantum mechanics. We show that they are related by the Fourier transform, more specifically we show that a momentum wave function given by the Fourier transform of the space wave function satisfies the requirements to be a momentum wave function, that is, normalization (the integral of its magnitude square equals one) and gives the correct formula for the expected value of the momentum.

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The momentum in quantum mechanics is defined as the operator


(this is in one variable, and p=-i*hbar*Gradient for more variables). The expected value of p, denoted by <p> is then

<p>=int_{-infinity}^{infinity} psi^*(x) (hbar/i) (d psi/dx)(x)dx,

and this is in space representation.

In the momentum representation, we look for g(p) that satisfies (15.60) and (15.61). As it is claimed there in the attachment, such g can be taken to be the Fourier transform of psi (with the hbar normalization in the definition of Fourier ...

Solution Summary

The solution discusses the momentum representation and the relationship between space wave function and momentum wave function.

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Vibrational frequency, Earth mass, harmonic function, hydrogen atom degeneracy

See attached file for full problem description and clarity in symbols.

1. a) In the infrared spectrum of H79Br, there is an intense line at 2630 cm¡1. Calculated the force constant
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potential when it has the wavefunction:
i) y(f ) = r1
p sin3f
ii) y(f ) = r 1
2p e¡3if :

2. The Earth (mass »=6£1024 kg) rotates about the sun (at an average distance of r »=1:5£1011 m) once
per year (one year = 10p million seconds).
a) What is the Earth's angular momentum with respect to the sun in units of ¯h?
b) What is the smallest principle quantum number n that allows the electron in a hydrogen atom to
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c) How large is such an atom? Use h&#136;ri for the H atom in the 1s state to express this size.

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Y2;§2(q ;f ) =
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, yr
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r . Hint: Remember that
e§if = cosf §i sinf
Chem 120A, Spring 2006 1
d) Show that the smplitude,
y(q ;f ) =
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4. In class you were shown that not only are the 2l +1 m states of each l degenerate for the hydrogen
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c) What is hSyi for each of these states?

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