- #1

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