# My afjafjafja Quantization

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Page 1 Chemistry 101 - SEAS Quantization alá Schrödinger: A Summary of Some Exactly Soluble Systems (Time-Independent) S chrödinger W ave E quation ( SWE ) - our “new” reality: - h 2 8 !  2 m  2   + V   = E   ; where: ! 2  = # 2 # x 2  + # 2 # y 2  + # 2 # z 2  and: V = potential energy function. If it is only a function of position, then it has the same 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) = 14   \$ 0  ã q 1  q 2 r = K q 1  q 2 r ; where: q 1  & q 2  are the two point charges (in C). r is the distance between the charges q 1  & q 2  . K = Coulomb ! s Law constant = 14   \$ 0   #  9.0 x 10 9  newton(N)ãmeter(m) 2 coulomb(C) 2  . Recall: Nãm = joule (J). E = total energy of the system. If it is a bound system - such as the electrons 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. | | 2  is the probability density for observing the particle(s) described by  to be at the location (x,y,z).  Page 2 In the equations below, “h” = Planck ! s constant = 6.626 x 10 - 34  Jãs, “m” = mass of the particle.  Particle-in-a-Box (PIB): ã 1-Dimension PIB V in SWE:  V = 0 if: 0 \$   x \$   L ; otherwise V = %  .  Energy :   E n  = n 2 ãh 2 8ãmãL 2  Wave function: n (x) = 2L ãsin & '( ) * +  nã   ãxL n = 1, 2, 3, … ; L = length of “box”  ã 2-Dimensions PIB V in SWE:  V = 0 if: 0 \$   x \$   L x  ; 0 \$   y \$   L y  ; otherwise V = %  .  Energy : E nx ny  = h 2 8ãm ã & '( ) * +  n 2x L 2x  + n 2y L 2y   Wave function:   nx ny (x,y) = 4 L x ã L y  ãsin & '( ) * +  n x ã   ãxL x  ã sin & '( ) * +  n y ã   ãyL y  L x  = length of “box” in x-direction ; n x  = quantum # in x-direction = 1, 2, 3, …. L y  = length of “box” in y-direction ; n y  = quantum # in y-direction = 1, 2, 3, …. ã 3-Dimensions PIB V in SWE:  V = 0 if: 0 \$   x \$   L x  ; 0 \$   y \$   L y  ; 0 \$   z \$   L z  ; otherwise V = %  . Energy : E nx ny nz  = h 2 8ãm ã & '( ) * +  n 2x L 2x  + n 2y L 2y  + n 2z L 2z  Wave function: nx ny nz (x,y,z) = 8L x ã L y  ã L z  ã sin & '( ) * +  n x ã   ãxL x  ã sin & '( ) * +  n y ã   ãyL y  ã sin & '( ) * +  n z ã   ãzL z   L x  = length of “box” in x-direction ; n x  = quantum # in x-direction = 1, 2, 3, …. L y  = length of “box” in y-direction ; n y  = quantum # in y-direction = 1, 2, 3, …. L z  = length of “box” in z-direction ; n z  = quantum # in z-direction = 1, 2, 3, ….    Page 3 ã Hydrogenic atom (Zp + , 1 e-) V in SWE: V(r)  = K q 1  q 2 r = K (+Ze) ã ( -  e)r = -   K Ze 2 r  (Coulombic) Energy : E n  = -  Z 2 ãR y n 2  Wave function (orbital): n l  m l (r, , , - ) = R n l (r) ã Y l  m l ( , , - ) = R n l (r) ã f( , ) ã eiãm l ã -  ; i = - 1 .   [ SEE TABLE 12.1, page 549 of Zumdahl 6/e for specific orbitals.. ] R y  = Rydberg constant = 2.18 x 10 - 18  J/atom = 13.61 eV/atom n = principal quantum # = 1, 2, 3, …. l  = angular momentum quantum # = 0, 1, 2, …, n -1   for each n. ( l  = 0 = “s”; l  = 1 = “p”; l  = 2 = “d”; l  = 3 = “f”; then, alphabetical) m l  = magnetic quantum # = 0, ± 1, ± 2,…, ± l for each l . Spherical Polar Coordinates: 0 \$   r \$    %  ; 0 \$    ,   \$       (180º) ; 0 \$    -   \$   2    (360º) . R n l (r) = radial part - depends upon (n, l ), the subshell of the electron. Y l  m l ( , , - ) = angular part = spherical harmonic function - depends upon l  & m l , of the electron.  Notes:   2-D “box”: nx ny (x,y) = 2L x  ã 2L y  ã sin & '( ) * +  n x ã   ãxL x  ã sin & '( ) * +  n y ã   ãyL y  …. or …. nx ny (x,y) = 4L x ã L y  ãsin & '( ) * +  n x ã   ãxL x  ã sin & '( ) * +  n y ã   ãyL y  …. or …. nx ny (x,y) = 4A ã sin & '( ) * +  n x ã   ãxL x  ã sin & '( ) * +  n y ã   ãyL y  Where: A = area of 2-D “box” = L x  ã L y  . 3-D “box”:   nx ny nz (x,y,z) = 2L x  ã 2L y  ã 2L y  ã sin & '( ) * +  n x ã   ãxL x  ã sin & '( ) * +  n y ã   ãyL y  ã sin & '( ) * +  n z ã   ãzL z  …. or …. nx ny nz (x,y,z) = 8L x ã L y  ã L z  ã sin & '( ) * +  n x ã   ãxL x  ã sin & '( ) * +  n y ã   ãyL y  ã sin & '( ) * +  n z ã   ãzL z  …. or …. nx ny nz (x,y,z) = 8V ã sin & '( ) * +  n x ã   ãxL x  ã sin & '( ) * +  n y ã   ãyL y  ã sin & '( ) * +  n z ã   ãzL z  Where: V = volume of 3-D “box” = L x  ã L y  ã L z  .

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