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jc09
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Homework Statement
How many bound states are there quantum mechanically ?
We are told to approach the problem semi classically.
Consider the Hamiltonian function
H : R 2n → R
(whose values are energies), and for E0 < E1 the set
{(p, x) ∈ R 2n |H(p, x) ∈ [E0 , E1 ]} ⊆ R 2n
,
which we assume to have the 2n-dimensional volume V (2n) . It is a fact that when-
ever V (2n) is finite, then there are only finitely many (distinguishable) quantum
mechanical states. More precisely, one has
V (2n) hn ≈ ♯{states of energy E ∈ [E0 , E1 ]},
where h = 2π. Moreover, strict equality holds provided the l.h.s. is an integer.
Asked to consider
asked to consider the Hamiltonian function
H(p, x) = p1^2 2m1 + p2^2 2m2 + 1 /2 m1 ω 1^2 x1^2 + 1/ 2 m2 ω 2^ 2 x2^2 ,
and to determine the approximate number of states of energy E
≤ Etotal .
Hint: This is the equation of an el lipsoid in 4-dimensional phase space with
coordinates (p1 , p2 , x1 , x2 ). The volume of the ellipsoid with radii a, b, c, d is abcd
times the volume of the 4-dimensional unit sphere
I'm stuck trying to find a starting point for the problem