2 Consider an electron that is initially far away from a proton and has no

# 2 consider an electron that is initially far away

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2. Consider an electron that is initially far away from a proton and has no kinetic energy. Considering the kinetic energy the electron gains by virtue of its Coulombic attraction to the proton, evaluate the wavelength of the electron when it is: (a) At a distance of 1 m from the proton. (b) At a distance of 0.5 × 10 -10 m from the proton (typical of the ground state of the hydrogen atom). Discuss your result. 3. A system is described by the Hamiltonian H = - d 2 / dx 2 - kx 2 (a) Show that for a certain value of a (an arbitrary constant) the function exp (- ax 2 ) is an eigenfunction of H . This means that acting on the function with H yields the very same function multiplied by a number, which is called the eigenvalue. (b) What is the eigenvalue? 4. Practice with solutions for the particle in a box: (a) For the particle (mass m ) in a box (length a ) of quantum number n=1,2,3, what is the probability of being in the middle third of the box, for these three values of n. Remember that the probability distribution is the square of the magnitude of the wavefunction, and integrate this over that part of the box. (b) Show that for a particle (mass m ) in a box (length a ) that any two of the eigenfunctions for different energy levels satisfy the following relation: ψ 1 x ( ) 0 a ψ 2 x ( ) dx = 0 . When this integral, called the overlap integral, is zero, the two functions are said to be orthogonal. Orthogonality turns out to be a general property of the energy eigenfunctions in any quantum problem.

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5.
• Summer '16
• Alistair Sinclair

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