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# Exercises

1. According to the Larmor formula of classical physics, a non-relativistic electron whose instantaneous acceleration is of magnitude radiates electromagnetic energy at the rate

where is the magnitude of the electron charge, the permittivity of the vacuum, and the velocity of light in vacuum [49]. Consider a classical electron in a circular orbit of radius around a proton. Demonstrate that the radiated energy would cause the orbital radius to decrease in time according to

where is the Bohr radius, the electron mass, the reduced Planck constant, and

Here, is the fine structure constant. Deduce that the classical lifetime of a ground-state electron in a hydrogen atom is .

2. Let the , for , be a set of orthonormal kets that span an -dimensional ket space. By orthonormal, we mean that the kets are mutually orthogonal, and have unit norms, so that

for . Show that

3. Demonstrate that

in a finite-dimensional ket space.

4. Demonstrate that in a finite-dimensional ket space:
Here, , , are general operators.

5. If , are Hermitian operators then demonstrate that is only Hermitian provided and commute. In addition, show that is Hermitian, where is a positive integer. [53]

6. Let be a general operator. Show that , , and are Hermitian operators. [53]

7. Let be an Hermitian operator. Demonstrate that the Hermitian conjugate of the operator is . [53]

8. Suppose that and are two commuting operators. Demonstrate that

9. Let the be the normalized eigenkets of an observable , whose corresponding eigenvalues, , are discrete. Demonstrate that

where the sum is over all eigenvalues.

10. Let the , where , and , be a set of degenerate unnormalized eigenkets of some observable . Suppose that the are not mutually orthogonal. Demonstrate that a set of mutually orthogonal (but unnormalized) degenerate eigenkets, , for , can be constructed as follows:

This process is known as Gram-Schmidt orthogonalization.

11. Demonstrate that the expectation value of an Hermitian operator is a real number. Show that the expectation value of an anti-hermitian operator is an imaginary number.

12. Let be an Hermitian operator. Demonstrate that .

13. Consider an Hermitian operator, , that has the property that . What are the eigenvalues of ? What are the eigenvalues if is not restricted to being Hermitian? [53]

14. An operator is said to be unitary if

Show that if and then . [53]

15. Show that if is Hermitian then is unitary. [53]

16. Show that if the , for , form a complete orthonormal set, so that

then the , for , where is unitary, are also orthonormal. [53]

17. The eigenstates of some operator acting in an -dimensional ket space are written

for , where all of the are real. Suppose that the are orthonormal, and span the ket space. Deduce that is Hermitian.

18. Let be an observable whose eigenvalues, , lie in a continuous range. Let the , where

be the corresponding eigenkets. Demonstrate that

where the integral is over the whole range of eigenvalues.

19. Show that

where is a Dirac delta function, and a constant.

Next: Position and Momentum Up: Fundamental Concepts Previous: Continuous Spectra
Richard Fitzpatrick 2016-01-22