the time dependent schrodinger equation: can be “separated” to get the time- independent...
Post on 20-Dec-2015
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)()(),(
)(
txtx
eee tiikxtkxi
φψ
ωω
=Ψ⇒= −−
The time dependent Schrodinger equation:
can be “separated” to get the time-independent Schrodinger equation
which can be used to find the “stationary states” or standing waves in a potential.
“Laplacian”
Can we use our previous knowledge to guess some of the characteristics of a particle in a 3 dimensional “box”?
What are the boundary conditions?
What is the form of the wave function?
Can you deduce anything about the ground state? Higher states?
( )2322
212
2
321 2: nnn
mL
hELLLif ++===
Leads to “degenerate” states: unique states with the same energy!
U=0 inside the box
⎥⎦
⎤⎢⎣
⎡
∂∂
+∂∂
+∂∂
+∂∂
⎟⎠
⎞⎜⎝
⎛+∂∂
=∇2
2
22
2
22
22
sin
1
sin
cos12
φθθθ
θ
θrrrr
φθφθ
θ
sinsin
cossin
cos
rz
ry
rx
=
=
=conversion from cartesian coordinates to
spherical polar coordinates
Laplacian in spherical polar coordinates:
ErUd
d
rmd
d
d
d
rmdr
dR
rdr
Rd
mR=+
ΦΘ
−⎥⎦
⎤⎢⎣
⎡ Θ+
ΘΘ
−⎥⎦
⎤⎢⎣
⎡+
−)(
sin1
2cot
12
22 2
2
22
2
2
2
2
2
2
22
φθθθ
θhhh
The Schrodinger equation in spherical polar coordinates:
)()()()(where φθψ ΦΘ= rRr
Θ+−=Θ−Θ
+Θ
)1(sin
1
sin
cos2
22
2
lll θθθθ
θm
dd
dd
The polar part of the Schrodinger equation is:
With some rearrangement, this can be recognized as the associated Legendre equation:
Luckily, someone has already solved this equation, so we don’t have to: