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# 1.P Exercises

1. According to 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 electron mass. 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 reduced Planck constant, and

Here, is the velocity of light in a vacuum, and the fine structure constant. Deduce that the classical lifetime of a hydrogen atom is .

2. Demonstrate that

in a finite dimensional ket space.

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

1.

2.

3.

4.

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

5. Let be a general operator. Show that , , and are Hermitian operators.

6. Let be an Hermitian operator. Demonstrate that the Hermitian conjugate of the operator is .

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

where the sum is over all eigenvalues, and denotes the unity operator.

8. Let the , where , and , be a set of degenerate 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.

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

10. Let be an Hermitian operator. Demonstrate that .

11. Consider an Hermitian operator, , that has the property that , where is the unity operator. What are the eigenvalues of ? What are the eigenvalues if is not restricted to being Hermitian?

12. 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, and denotes the unity operator.