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EPL202: Problem Set 1 1. Consider a thermal neutron, that is, a neutron with speed v corresponding to average thermal energy at the temperature T=300K. Is it possible to observe a diffraction pattern when the beam of such neutrons fall on a crystal? (b) In a large accelerator, an electron can be provided with energy over 1 GeV=109 eV. What is the de Broglie wavelength corresponding to such electrons. 2. Consider the following wavefunction for a quantum mechanical particle (say electron) ( x) Ae (a) (b) ( x x0 ) 2 4l 2 .x From the normalization condition find out the constant A , Why is this normalization condition necessary. Find out the average position of this particle, namely x . What is the physical meaning of x . (c) (d) Find out the average momentum of this particle , p x .The momentum operator can be introduced as i x Calculate the following quantities, namely x ( x x) 2 & p x ( p x p x ) 2 and verify that . Interpret this result physically. 2 (e) Repeat all the calculations for a plane wave, e ikx and list all the difficulties you run. 3. Consider the following general form of the wave-fucntion which has 1 been introduced in the class ( x) a(k )e ikx dk . Show that the 2 average momentum associated with such a particle is given by p x dp x p x | a(k ) | 2 . Interpret this result. p x x 1 0 0 4. Let us consider the following operator matrix 0 0 1 . Find out 0 1 0 the eigenvalues and the eigenfunctions of this operator. Is this operator Hermitian? 5. Prove the following properties of a hermitian operator. (a) A hermitian operators has real eigenvalues. (b) Eigenvectors of hermitian operator with distinct eigenvalues are orthogonal. 6. Write down the operators used for the following quantities in quantum 2 mechanics x, p x , p x , p, Lz , L2 . Check if they are hermitian or not. , xˆ ipˆ x 7. Check if the following operators are hermitian x 8. A wave function is given by ( x) Ae ( x x0 ) 2 4l 2 . Find out its fourier transform. Interpret this result in terms of uncertainty in position and momentum. ( very similar to the first problem). 9. Write down the definition of the Dirac function. Can the wave function of an electron be given by a Dirac function. Write down the fourier transformation of ( x) ( x) . What does it mean?