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Chemistry 101 - SEASQuantization al Schrdinger:

A Summary of Some Exactly Soluble Systems

(Time-Independent) Schrdinger Wave Equation (SWE) - our new reality:

-h2

8!2m 2 + V = E ;

where: !2" =#2"#x2

+#2"#y2

+#2"#z2

and:

V = potential energy function. If it is only a function of position, then it has thesame form as its classically expected version (Coulomb!s Law, etc.).

For example: If two charges - that can be considered point charges- are within an atom or within a molecule, then the pair-wise interaction is:

V = V(r) =1

4"$0

q1q2r

=K q1q2

r ; where:

q1 & q2 are the two point charges (in C).

r is the distance between the charges q1 & q2.

K = Coulomb!s Law constant

=1

4"$0 # 9.0 x 109

newton(N)meter(m)2

coulomb(C)2.

Recall: Nm = joule (J).

E = total energy of the system. If it is a bound system - such as theelectrons in an atom or in a molecule - then the energy is quantized

and will depend upon one or more quantum number(s).

" = "(x,y,z) [in general] = Wave function, a.k.a. probability amplitude.|"|2is the probability density for observing

the particle(s) described by "to be at thelocation (x,y,z).

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In the equations below, h = Planck!s constant = 6.626 x 10-34Js,

m = mass of the particle.Particle-in-a-Box (PIB): 1-Dimension PIB

V in SWE: V = 0 if: 0 $x $L ; otherwise V = %.

Energy: En =

n2h2

8mL2

Wave function: "n(x) =2

Lsin&'

()*+n"x

L

n = 1, 2, 3, ; L = length of box

2-Dimensions PIBV in SWE: V = 0 if: 0 $x $Lx ; 0 $y $Ly; otherwise V = %.

Energy: Enxny

=h2

8m

&'

(

)*

+n2xL2x

+n

2

y

L2y

Wave function: "nxny(x,y) =4

Lx Lysin

&'(

)*+nx"x

Lx sin

&'(

)*+ny"y

Ly

Lx= length of box in x-direction ;

nx= quantum # in x-direction = 1, 2, 3, .

Ly= length of box in y-direction ;

ny= quantum # in y-direction = 1, 2, 3, .

3-Dimensions PIBV in SWE: V = 0 if: 0 $x $Lx ; 0 $y $Ly; 0 $z $Lz; otherwise V = %.

Energy: Enxnynz =

h2

8m

&'(

)*+n2x

L2

x

+n

2

y

L2

y

+n

2

z

L2

z

Wave function:

"nxnynz(x,y,z) =8

Lx Ly Lz sin

&'(

)*+nx"x

Lx sin

&'(

)*+ny"y

Ly sin

&'(

)*+nz"z

Lz

Lx= length of box in x-direction ;nx= quantum # in x-direction = 1, 2, 3, .

Ly= length of box in y-direction ;

ny= quantum # in y-direction = 1, 2, 3, .

Lz= length of box in z-direction ;

nz= quantum # in z-direction = 1, 2, 3, .

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Hydrogenic atom (Zp+, 1 e-)

V in SWE: V(r) =K q1q2

r =

K (+Ze) (-e)

r = -

K Ze2

r (Coulombic)

Energy: En = -Z2Ry

n2

Wave function (orbital):"n lml(r,,,-) = Rn l(r) Ylml(,,-) = Rn l(r) f(,) e

iml- ; i = -1 .

[SEE TABLE 12.1, page 549 of Zumdahl 6/e for specific orbitals.. ]

Ry= Rydberg constant = 2.18 x 10-18J/atom = 13.61 eV/atom

n = principal quantum # = 1, 2, 3, .l= angular momentum quantum # = 0, 1, 2, , n -1for each n.

(l= 0 = s; l= 1 = p; l= 2 = d; l= 3 = f; then, alphabetical)

ml= magnetic quantum # = 0, 1, 2,,l for each l.

Spherical Polar Coordinates: 0 $r $% ; 0 $,$"(180) ; 0 $-$2"(360) .

Rn l(r) = radial part - depends upon (n,l

), the subshell of the electron.Ylml

(,,-) = angular part = spherical harmonic function

- depends upon l& ml, of the electron.

Notes:2-D box:

"nxny(x,y) =2

Lx

2

Ly sin

&'(

)*+nx"x

Lx sin

&'(

)*+ny"y

Ly . or .

"nxny(x,y) =4

Lx L

y

sin

&

'(

)

*+nx"x

Lx

sin

&

'(

)

*+ny"y

Ly

. or .

"nxny(x,y) =4

A sin

&'(

)*+nx"x

Lx sin

&'(

)*+ny"y

Ly

Where: A = area of 2-D box = Lx Ly.

3-D box:

"nxnynz(x,y,z) =2

Lx

2

Ly

2

Ly sin

&'(

)*+nx"x

Lx sin

&'(

)*+ny"y

Ly sin

&'(

)*+nz"z

Lz

. or .

"nxnynz(x,y,z) =8

Lx Ly Lz sin

&'(

)*+

nx"x

Lx sin

&'(

)*+

ny"y

Ly sin

&'(

)*+

nz"z

Lz

. or .

"nxnynz(x,y,z) =8

V sin

&'(

)*+nx"x

Lx sin

&'(

)*+ny"y

Ly sin

&'(

)*+nz"z

Lz

Where: V = volume of 3-D box = Lx Ly Lz.