CH110 Study Guide - Final Guide: Eigenfunction, Angular Velocity, Standard-Definition Television
eigenfunction relevant to the quantum numbers n = 1 and l = m = 0,
whereas for ψ2(r) the eigenfunction relevant to n = 2 and l = m = 0.
Then (2.1) yields
1 1it −r a2 r−1r−3it Ψ(r,t)=e2a1e+221−2e2 8, (2.18)
in which the coefficients a1 and a2 supposed to be real for simplicity’s
sake are related by a12 + a2 = 1 in view of (2.2). If need be, Eq. (2.18)
permits one to find Ψ(r,0).
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The probability density of the coordinate r is
2 1 −r 2 −r a2 r2 a1a2 r −1r
(r,t)≡|Ψ(r,t)| = e a1e + 8 1−2 + 2 1−2e 2 cos83t,
(2.19)
from which it is seen that the atom pulsates remaining spherically
symmetric. The mean energy of the atom can be found directly from
(2.4):
E = − 1
2 + 8
3 a 2 2 , ( 2 . 2 0 )
where the coefficient a1 is excluded and 0 ≤ a2 ≤ 1. One sees that, in the
case of the atom, the mean energy varies continuously as well, from E1 =
−1/2 to E2 = −1/8 in this example.
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Document Summary
The probability density of the coordinate r is eigenfunction relevant to the quantum numbers n = 1 and l = m = 0, in which the coefficients a1 and a2 supposed to be real for simplicity"s. If need be, eq. (2. 18) sake are related by a1. 2 + a2 whereas for 2(r) the eigenfunction relevant to n = 2 and l = m = 0. 22(cid:1736)(cid:1737)1 2(cid:1739)(cid:1740)e2 8(cid:1745)(cid:1746), (2. 18) permits one to find (r,0). 2 1 r(cid:1741) 2 r a2(cid:1735) r(cid:1738)2 a1a2(cid:1735) r(cid:1738) 1r (cid:1744) 8 (cid:1736)(cid:1737)1 2(cid:1739)(cid:1740) (cid:591)(r,t) | (r,t)| = (cid:590)e (cid:1742)a1e + (2. 19) (cid:1743)(cid:1746) E = 1 where the coefficient a1 is excluded and 0 a2 1. 1/2 to e2 = 1/8 in this example. from which it is seen that the atom pulsates remaining spherically symmetric. The mean energy of the atom can be found directly from (2. 4):
