HW #6

1. Free Particle Propagator

(a)

To simplify the notation, we write t = t'' - t', x = x'' - x'.  

First of all, because the base kets evolve according to the "wrong sign" Schrödinger equation (see pp. 87-89),
|x', 
            t'⟩ = 
          (e^(+i H t'/ℏ)) | x', 
        0⟩, \(TraditionalForm\`\(⟨\(x\*""", 
              t\)\*""" | \) = ⟨\(x\*""", 
            0 | \(\(e\^\(\(-i\)\ H\ \(t\*"""/ℏ\)\)\)\(.\)\)\)\) 

Therefore,
\(TraditionalForm\`⟨x'', t'' | x', 
          t'⟩ = \(⟨x'' | 
              e\^\(\(-i\)\ H\ \((t'' - t')\)/ℏ\) | 
              x'⟩ = ∫d\ p ⟨x'' | p⟩ ⟨p | e\^\(\(-i\)\ H\ t/ℏ\) | 
                  x'⟩\)\)
\(TraditionalForm\`\(\(=\)\(∫d\ p\ \(e\^\(i\ p\ x''/ℏ\)\/\((2  π\ ℏ)\)\^\(1/2\)\) ⟨p | 
                e\^\(\(-i\)\ \((p\^2/2  m)\)\ t/ℏ\) | 
                x'⟩\)\)\)
\(TraditionalForm\`\(\(=\)\(∫d\ p\ \(e\^\(i\ p\ x''/ℏ\)\/\((2  π\ ℏ)\)\^\(1/2\)\) \(e\^\(\(-i\)\ \((p\^2/2  
                      m)\)\ t/ℏ\)\) 
            e\^\(\(-i\)\ p\ x'/ℏ\)\/\((2  π\ ℏ)\)\^\(1/2\)\)\)\)
\(TraditionalForm\`\(\(=\)\(∫d\ p\ \(e\^\(i\ p\ x/ℏ\)\/\(2  π\ ℏ\)\) e\^\(\(-i\)\ p\^2\ t/2\ m\ ℏ\)\)\)\)
\(TraditionalForm\`\(\(=\)\(∫d\ p\ \(1\/\(2  π\ ℏ\)\) \(exp(\(-i\)\ \(t\/\(2\ m\ ℏ\)\) \((p - \(m\ x\)\/t)\)\^2 + 
                i\ \(m\ x\^2\)\/\(2  ℏ\ t\))\)\)\)\)
\(TraditionalForm\`\(\(=\)\(\(1\/\(2  π\ ℏ\)\) \(\@\(\(2  π\ m\ ℏ\)\/\(i\ t\)\)\) e\^\(i\ m\ x\^2/2  ℏ\ t\)\)\)\)
\(TraditionalForm\`\(\(=\)\(\(\@\(\(\(m\)\(\ \)\)\/\(2\ π\ i\ ℏ\ t\)\)\) \(\(e\^\(i\ m\ x\^2/2  ℏ\ t\)\)\(.\)\)\)\)\)

(b)

For the classical trajectory, the velocity is simply v = (x/t), and hence the action is (Subscript[S, c]) = ∫(1/2) 
              m (((Overscript[x, ·]))^2) d t = (1/2) 
              m ((((x/t)))^2) t = ((m (x^2))/(2 t)), and hence the exponent of the propagator is indeed (i (Subscript[S, c])/ℏ).

Geometric Optics

(a)

Following the same steps we did for the Schrödinger equation, we first write the Maxwell's equation with (A^0) = (e^(i S/ℏ)):
\(\((\(n\^2\/c\^2\) ∂\^2\/∂\ t\^2 - \(∇\& -> \)\^2)\) e\^\(i\ S/ℏ\)\)=(\(\(n\^2\/c\^2\) \((\(i\ S\&\(\(·\)\(·\)\)\)\/ℏ + \((\(i\ \(S\&·\)\)\/ℏ)\)\^2)\)\)-((i (((Overscript[∇,  ->]) )^2) S)/ℏ)-(((i (Overscript[∇,  ->]) S)/ℏ))2)(e^(i S/ℏ))=0.
In the limit S ≫ ℏ, we can drop terms of O(S/ℏ) and keep those of ((O(S/ℏ))^2):
	((n^2)/(c^2)) (((Overscript[S, ·]))^2)-((Overscript[∇,  ->]) S)2=0.
This is the "Hamilton-Jacobi" equation.

(Of course, in the Maxwell equation, there is no notion of ℏ.  What we are doing is a valid approximation when the variation of the phase is very fast compared to the variation of the index of refraction.  It is called "eikonal approximation" in optics.)

(b)

Assuming n((Overscript[x,  ->]) ) = n(x), the "Hamilton-Jacobi" equation does not have an explicit dependence on y or t.  Writing
	S(x, y, 
          t) = (Overscript[S,  ∼]) (x, (Subscript[p, y]), E) + (Subscript[p, y]) y - E t,
the equation becomes
	((n^2)/(c^2)) (E^2)-((d/(d x)) (Overscript[S,  ∼]) )2-((Subscript[p, y])^2)=0.
Therefore,
	\(TraditionalForm\`\(d\ \(S\& ∼ \)\)\/\(d\ x\) = \@\(\(n\^2\/c\^2\) E\^2 - p\_y\^2\)\)
and hence
	\(TraditionalForm\`\(S\& ∼ \) = ∫\^x\(\@\(\(\( n(x')\)\^2\/c\^2\) E\^2 - p\_y\^2\)\) d\ x'\).
Here, the possible x-dependence of the index of refraction is emphasized.

(c)

Using \(TraditionalForm\`t = ∂\(\(\(\ \)\(S\)\)\& ∼ \)\/∂\ E\), \(TraditionalForm\`y = \(-\(∂\(\(\(\ \)\(S\)\)\& ∼ \)\/∂\ p\_y\)\)\), we find
	t =  (∫^x)((((( n(x'))^2)/(c^2)) 
                  E)/(Sqrt[(((n(x'))^2)/(c^2)) (E^2) - ((Subscript[p, y])^2)])) 
            d x',
	\(TraditionalForm\`y = ∫\^x\ \(p\_y\/\@\(\(\(n(x')\)\^2\/c\^2\) E\^2 - p\_y\^2\)\) d\ x'\).

(d)

Assuming that n(x) = (Subscript[n, 1]) for x < 0 and n(x) = (Subscript[n, 2]) for x > 0, and choosing the lower end of the integration at x = 0,
	\(TraditionalForm\`\(y\_ < \)(
          x) = \ \(p\_y\/\@\(\(n\_1\^2\/c\^2\) E\^2 - p\_y\^2\)\) x\) for x < 0
and
	\(TraditionalForm\`\(y\_ > \)(
          x) = \ \(p\_y\/\@\(\(n\_2\^2\/c\^2\) E\^2 - p\_y\^2\)\) x\) for x > 0.
Using the trigonometric relation sin α = 
        (1/(Sqrt[(cot^2) α + 1])), we find
	\(TraditionalForm\`sin\ \(α\_ < \) = \(1\/\@\(\(\(n\_1\^2\/c\^2\) E\^2 - p\_y\^2\)\/p\_y\^\(\(2\)\(\ \)\) + 1\) = \(c\ p\_y\)\/\(\(n\_1\) E\)\)\),
	\(TraditionalForm\`sin\ \(α\_ > \) = \(1\/\@\(\(\(n\_2\^2\/c\^2\) E\^2 - p\_y\^2\)\/p\_y\^\(\(2\)\(\ \)\) + 1\) = \(c\ p\_y\)\/\(\(n\_2\) E\)\)\),
and hence
	\(TraditionalForm\`\(sin\ \(α\_ < \)\)\/\(sin\ \(α\_ > \)\) = n\_2\/n\_1\),
which is nothing but the Snell's law of refraction.

Bohr-Sommerfeld

We use Eq. (35) of the lecture notes
(∫_((Subscript[x, min]))^((Subscript[x, max])))(Sqrt[2  m(
                    E - V(x))]) d x = (n + (1/2)) π ℏ,
where (Subscript[x, min])and (Subscript[x, max])are the classical turning points.  For a harmonic oscillator, V(x) = (1/2) m (ω^2) (x^2), and hence -(Subscript[x, min]) = (Subscript[x, max]) = (Sqrt[((2  
                      E)/(m (ω^2)))]).  The integral is

Integrate[m ω (Sqrt[ (X^2) - (x^2)]), {x, -X, X}]

PowerExpand[%]

Here, we used X = (Subscript[x, max]).  Therefore,
	(1/2) 
          m π ((2 E)/(m (ω^2))) ω = (n + 
              (1/2)) π ℏ,
and hence
	E = (n + (1/2)) ℏ ω.
This result agrees with the exact result, which is an accident for this particular case.