2.6. Problems¶
2.1.¶
The product of the exponentials of two noncommuting operators and can be written as This is the socalled Campbell Baker Hausdorff formula (or CBH formula). The aim of this problem is to derive this formula.

First, show that .

Now expand the formula to second order in and to derive the CBH formula.

If you're brave, you may try to find the third order expansion to find the next term.
2.2.¶
We consider a particle moving in one dimension. The state of the particle is .

Show that by inserting a unit operator using completeness of the momentum basis .

Demonstrate the same result by writing as the operator and Taylor expanding around .

Show that for a wave function in three dimensions: where is the vector rotated about an angle around the axis.

(The following two parts were already addressed in problem 1.9). Show that the operator rotates a spin1/2 state about an angle around the axis.

Give the rotation operator for the wave function describing a spin1/2 particle.
2.3.¶
The Lagrangian of a harmonic oscillator is . Show that the classical action is: where
Hint: the definition of the action is: . Make use of a classical trajectory of the harmonic oscillator: , which satisfies boundary conditions: and .
2.4.¶
The path integral formalism expresses the probability to move from a point at time to a point at time in terms of a sum over all paths: We write the path as a sum of the classical path and a fluctuation: where , since the positions at the beginning and at the end of the path are fixed.
We consider a free particle.

Give the action for the classical path for a free particle (that is, for all ).

Show that the action for a general path can be written as Also show that

The properly normalised form of the discretised path integral is given by In the exponent, means ; . Furthermore, and .
The integration over all possible paths can be performed in the discretised path integral: It can be shown (using e.g. induction) that If you're in for a challenge, you may try to prove this, but that is not required here.
Show, using this result, that

Find this propagator directly, using Hint: insert two unit operators in this expression, formulated as integrals over .