MAT 720Homework 21. (a) Show by explicit integration that = 0 for the ground state of a harmonic oscillator. V (x) = kx2(b) Show by explicit integration that = 0 for the ground state of a harmonic oscillator.(c) Show that the Heisenbergs uncertainty principle is not violated for the ground state.

2. Problem 2.51 (Griffiths 2nd edition).

3. Calculate the frequency of radiation needed to excite an electron trapped in an infinite well from its ground state to its second excited state. The width of the well is 10 nm.

4. Derive the uncertainty relationship between energy and time. (Do not hand this in)

5. Show that L x L = iL, where L is the angular momentum operator.

6. (a) Study the hermiticity of these operators: x, d/dx, and i(d/dx). What about the complex conjugate of these operators? Are the Hermitian conjugates of position and momentum operators equal to their complex conjugates?

(b)Use the results in (a) to discuss the hermiticity of the operators exp (x), exp (d/dx), and exp(id(dx))(c) Find the Hemitian conjugate of xd/dx.(d) Use the results above to discuss the hermiticity of the components of the angular momentum operator (Lx, Ly and Lz)

7. (a) Find the eigenvalues and eigenfunctions of the operator . Restrict you for the eigenfunctions to those complex function that vanish everywhere except in the region 0 < x < a.(b) Normalize the eigenfunction and find the probability in the region 0 < x < a/2. Do you expect the result you obtained?

8. Read the discussion on particle in an infinite well in Griffith. Please read the entire section and also look at Example 2.2.

9. In class we looked at the case for a particle with energy E < V0 being incident of a potential barrier of height V0 and width a. Work out the same problem for the case when E > V0. (Do not hand this in).