Born interpretation of the wavefunction

d²R 2 dR 2"
Radial wave equation —7 + + -V
^M dr² r dr h¹

e<sup>2

E</sup> + 4^o
has many solutions, one for each positive integer of n.
o Solutions are of the form (see Appendix N of Eisberg & Resnick):

Rnl(r) = e-^Zr,"^a"	’Zr'	Gnl	uJ

where a₍₎ is the Bohr radius. Bound-state solutions aie only acceptable if _E _ 〜
" (,47T£₀)²2h²n^2
ev
where n is the principal quantum number, defined by /?=/+/,/ +2, I +3,...
o E_n only depends on n: all I states for a given n are degenerate (i.e. have the same energy).
What is the ground state of hydrogen (Z=l)? Assuming that the ground state has n = I,I = 0 Eqn. 6 can be written

r² dr\ dr)

+

/? = ()
o Taking the derivative
Try solution R = where A and a₀ are constants. Sub into Eqn. 7: 2/^²




expression to zero =>
Setting first term to zero =>

a

£=—^■ = -13.6 eV

r Same as Bohr’s results
To satisfy this Eqn. for any r,both expressions in brackets must equal zero. Setting the second
Using the Equivalence Principle, the classical dynamical quantities can be replaced with their associated differential operators:
Substituting, we obtain the operator equation:
Assuming electron can be described by a wavefunction of form, =


can write

or


is the Laplacian operator.
where.
o Since V(x,y_tz,) does not depend on time, = y\x₉y,z)e ^,El,h is a solution to the
Schrodinger equation and the eigenfunction is a solution of the time-independent
Schrodinger equation'.
▽ V(x,y,z) + = Ey/(x,y,z)
o As V = V(r), convenient to use spherical polar coordinates.

where

dfj)¹

Can now use separation of variables to split the partial differential equation into a set of ordinary differential equations.

Born interpretation of the wavefunction

o Principle of QM: the wavefunction contains all the dynamical information about the system it describes.
Born interpretation of the wavefunction: The probability (P(x,t)) of finding a particle at a position between x and x-\-dx is proportional to W(x,t)\²dx\
P(x,t) = n,t) ^V(xJ) = W(xj)\²
P(x,t) is the probability density.
Immediately implies that sign of wavefunction has no
direct physical significance.
Radial solutions (R( r ))
o The radial probability function P_nl(r), is the probability that the electron is found between r

Separation of the Schrodinger equation
As the LHS of Eqn 3 does nor depend on r or 0 and RHS does not depend on(j)their
common value cannot depend on any of these variables.

and RHS becomes 1 d	dR'	1 d
Rdr	dr/	®s\nOdO

Setting the LHS of Eqn 3 to a constant:

1

dO

- V(r)] = 1 sin^—
ti² sin~ 0 Qsin0d0\ d0)

2
sin²0

_ \_d_
◊a

1 d sind d0\^n d0)
1 d r^2 dr	^(r2 dR' dr)

o Both sides must equal a constant, which we choose as 1(1+1):

(5)
(6)


o We have now separated the time-independent Schrodinger equation into three
ordinary differential equations, which each only depend on one of ^(4), &(5) and R(6).
Separation of the Schrodinger equation
o Assuming the eigenfunction is separable: (2)
o Using the Laplacian, and substituting (2) and (I):
+ V(r)/?0O=E/?0O
o Carrying out the differentiations,

R


'i² 0
d


r²sin0<^^^nd7e}


RQ d²d>


,, ,+V(/-)/J0
r² sin² J


o Note total derivatives now used, as 7? is a function of r alone, etc.


o Now multiply through by -2/zr² sin² 0/RQlr and taking transpose,


(3)


I J²<1>


2A
tr


r²sinM£-V(r)]

1 0 (Z/_Qo)<2exp(-Zr/a_o)
2 0 (Z/2a₀)i2 (1 - exp(-Zr/2a₀)

2. 
   1 (Z/2_ao)^(^)exp(-Zr/2_ao)
   

3 0 (Z/3a_o)J2[l-(2Z"3ao) + §(e)^{1 2 3}] W"3
3 1 (Z/W_3)(^)(l-^)exp(-Zr/3«₀>
3 2 (Z/3a₀)f(2^/3y/5) (^)²_exp(-Zr/3a₀)

PY3P05