Quantum Chemistry 1 (CHEM 565) Fall 2018 Gerald Knizia Department of Chemistry The Pennsylvania State University Quantum Chemistry 1 (CHEM 565), Fall 2018 1 of 54
Quantum Chemistry 1 (CHEM 565)Fall 2018
Gerald KniziaDepartment of ChemistryThe Pennsylvania State University
Quantum Chemistry 1 (CHEM 565), Fall 2018
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Grand Topic #3
3. Simple Model Problems
Free particles & Wave packets (Plane wave solutions; Superposition principle; Gaussian
wave packets)
Numerical solution of 1-particle quantum problems (Dimensionless EOM; Real-space
grid discretization; Time evolution; Quantum phenomena (eigenstates, tunneling, Husimi-
repr.); Pseudo-spectral methods; Basis function decomposition/1)
Particle in a box & co (1D Case: Steps, barriers, wells; Transmission/reflection,
resonance, tunneling; 3D Case: Separation of variables)
Vibrations & Harmonic oscillator (H.-o. (1D,3D), Polyatomic case, Non-harmonic
oscillator (1D) & QM approximation methods (subspace projections, perturbation expansion,
variational method)), outlook: field quantization
Spherical potentials & Angular momenta (Separation of diatomic nuclear Schrödinger
equation; Rotational spectra; Ro-vibrational spectra )
Spin & Two-level systems (Single spin ½; Coupled spins; Combining orbital & spin-
degrees of freedom)
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3.2 Particles & boxes
Simple Model Problems—Particle in a box & co.
We will consider some piece-wise constant potentials („step potentials“). Why?
While not terribly realistic, these are analytically comparatively simple tohandle,
...and already show a variety of important quantum phenomena(in particular, transmission, reflection, and tunneling of wave packets)
Additionally, their eigenstates (plane waves and exponentials) are highlyrelevant as basis functions for expanding approximate wave functions inrealistic model systems (e.g., uniform electron gas, and structures with periodic
boundary conditions (polymer/slab/solids))
...and this is where the quantum version of the ideal gas lives( ensemble states ).↘ ρ̂
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3.2 Particles & boxes
Piecewise constant potentials—General solution approach for eigenstates
Divide real-space into regions of constant potential
Constant potential ansatz or , individually for eachsub-region
Stick sub-potential-solutions together by demanding that values of wavefunctions and derivatives of wave functions agree at boundaries.
(...unless boundary goes to a region, in which case at the boundary and no
condition on the derivative arises)
⇒ ψ( ) = Cx⃗ eı xki ψ( ) = Cx⃗ e xρi
i
V (x) = ∞ ψ(x) = 0
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3.2 Particles & boxes
Consider the following potential:
V (x) =⎧⎩⎨⎪⎪
∞ for x ≤ 00 for x ∈ (0,L)∞ for x ≥ L
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
Time-independent Schrödinger equation for 1 particle in positionrepresentation (where ):
Particle cannot be in (non-zero measure) region with infinite potential
The wave function needs to be continuous (earlier discussion of particle flux!). Inner function with ( ) has boundary conditions and
.
= −ıℏP̂ ∂x
(− + V (x))ψ(x) = Eψ(x)ℏ2
2md2
dx2
⇒ ψ(x)ψ(x)
==
0 for x ≤ 00 for x ≥ L
⇒ ψ(x) x ∈ [0,L] ψ(0) = 0ψ(L) = 0
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
Schrödinger equation for inner part of wave function:
with linear homogenous ordinary differential equation (ODE) with
constant coefficients
ψ(x) = Eψ(x)−ℏ2
2md2
dx2
ψ(x) + ψ(x) = 0d2
dx2
2mE
ℏ2
ψ“(x) + q ⋅ ψ(x) = 0
q = 2mE
ℏ2 →
ψ“(x) + qψ(x) = 0 (with q = )2mE
ℏ2
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
Ansatz: . Inserting this into the differential equation:
General solution: with
Almost identical to previously discussed free particle case, but...
ψ(x) = eıkx
ψ“(x) + qψ(x)(ık + q)2eıkx eıxk
((ık + q))2 eıkx
k2
⇒ k
====
=
000 (note: exp(x) is never 0)q
± = ±q√2mE
ℏ
− −−−−√⇒ ψ(x) = +C1e
ıkx C2e−ıkx k = 2mE
ℏ
− −−−√
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
This provides an infinity of different solutions ( and are free). However,not all of them fit to the boundary conditions and .
To see which ones do, it is helpful to reformulate the exponentials intotrigonometric functions with
E C1
ψ(0) = 0 ψ(L) = 0
exp(ıkx) = cos(kx) + ı sin(kx) (Euler formula).
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
Inserting this into the general solution for :
with , two new arbitrary constants (instead of and ). This used:
x ∈ [0,L]
ψ(x) =====:
+C1eıkx C2e
−ıkx
(cos(kx) + ı sin(kx)) + (cos(−kx) + ı sin(−kx))C1 C2
(cos(kx) + ı sin(kx)) + (cos(kx) − ı sin(kx))C1 C2
( + ) cos(kx) + ı( − ) sin(kx)C1 C2 C1 C2
A cos(kx) + B sin(kx)
A B C1 C2
cos(x) = cos(−x) and sin(−x) = − sin(x).
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
Since , we can instantly see that :
Thus, only functions of the form can potentially becompatible with .
ψ(x) = A cos(kx) + B sin(kx)
ψ(0) = 0 A = 0
0 = ψ(0) = A + B = Acos(0) =1
sin(0) =0
ψ(x) = B sin(kx)ψ(0) = 0
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
For to hold, for must be located at one of the zeros of theSinus function, which lie at ( ). Therefore:
produces (not normalizable). Furthermore, solutionsfor and are identical. So only need be considered.
ψ(L) = 0 kx x = L
n ⋅ π n ∈ Z
sin(kL) = 0⇒ kL = πn
⇒ L = nπ2mE
ℏ2
− −−−−√⇒ E = = (n ∈ Z)
n2π2ℏ2
2mL2
n2h2
8mL2
n = 0 ψ(x) = B sin(0) = 0n −n n ≥ 1
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
The free constant in can be fixed by demanding that(i.e., that the wave function is normalized).
The Schrödinger equation itself has an infinite, continuous range ofsolutions for any energy.
However, of those only a discrete subset is compatible with theboundary conditions of wave function continuity[*] and normalizability.
([*]: At points where is finite, the wave function also must be differentiable)
E = = (n = 1, 2, 3, …)n2π2ℏ2
2mL2
n2h2
8mL2
B ψ(x) = B sin(x)∫ |ψ(x) dx = 1|2
x V (x)
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
Comment 2: This is a general feature of bound states: Since
is a 2nd order ODE, once you know and at any point , you cancompute for all other , for any constant choice of
( initial value problem, Picard–Lindelöf theorem).
Quantization of the energy thus comes from the conditions on thephysicality of the so-computed wave functions (i.e., boundary conditions and
normalizability).
It is not a consequence of the Schrödinger equation.
(− + V (x) − E)ψ(x) = 0ℏ2
2md2
dx2
ψ(x) (x)ψ′ x
ψ(x) x E
→
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
Note that the lowest possible energy (the ground state energy) is not zero
Comment 3: Absolute energies are not observable, but differences are.
In many cases the form of the potential changes during a process, andthen differences in zero-point energy between the initial and final states canmatter.
(e.g., consider the vibrational ground states for a chemical reaction for
diatomics. The ground state energies of the inter-nuclear potentials , ,
, are not observable, but they add a generally non-zero contribution to the
reaction energy. This is experimentally well confirmed.)
E = = (n = 1, 2, 3, …)n2π2ℏ2
2mL2
n2h2
8mL2
V (x)
AB + CD → AC + BD
( )VAB RAB ( )VCD RCD
( )VAC RAC ( )VBD RBD
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
Comment 4: Note that the wave function lives entirely in a region ofvanishing potential (i.e., at all points where ). That meansthat all of the (non-zero) energy of the states is kinetic energy.
Furthermore, the kinetic energy scales as
That is, it becomes larger the more a particle is confined.
E = = (n = 1, 2, 3, …)n2π2ℏ2
2mL2
n2h2
8mL2
ψ(x)V (x) = 0 |ψ(x) ≠ 0|2
∝ .Ekin1L2
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3.2 Particles & boxes
Particle in a box—Standard boundary conditions
Comment 4: (cont'd) (...kinetic energy becomes larger the more a particle is confined)
This, too, is a general feature of quantum systems. It can be considered asa consequence of the Heisenberg uncertainty relation : reducingspace (reducing ) leads to an increase in momentum fluctuations ( ), and
thus kinetic energy ( ).
Lowering kinetic energy of electrons by spreading them over more space isa central mechanism in the formation of chemical bonds.
E = = (n = 1, 2, 3, …)n2π2ℏ2
2mL2
n2h2
8mL2
δx ⋅ δp ≥ ℏ/2δx δp
∝⟨ ⟩P 2
2m
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3.2 Particles & boxes
General step potentials
Schrödinger equation in region with constant potential:
Case 1, (i.e., classically allowed region):
Introduce positive constant such that
General solution to Schrödinger equation in this region:
where , are complex constants (fixed by boundary conditions)
( + V)ψ(x) = Eψ(x) ⇔ ψ(x) + ψ(x) = 0−ℏ2
2md2
dx2
d2
dx2
2m(E − V )
ℏ2
E > V
k
=k2 2m(E − V )
ℏ2
ψ(x) = +C1eıkx C2e
−ıkx
C1 C2
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3.2 Particles & boxes
General step potentials
Schrödinger equation in region with constant potential:
Case 2, (i.e., classically forbidden region):
Introduce positive constant such that
General solution to Schrödinger equation in this region:
where , are complex constants (fixed by boundary conditions)
( + V)ψ(x) = Eψ(x) ⇔ ψ(x) + ψ(x) = 0−ℏ2
2md2
dx2
d2
dx2
2m(E − V )
ℏ2
E < V
ρ
=ρ2 2m(V − E)
ℏ2
ψ(x) = +C1eρx C2e
−ρx
C1 C2
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3.2 Particles & boxes
General step potentials
In general: Write wave function like above in specific regions, and thenstitch them together using the boundary conditions at the region boundaries.
Let us consider some specific cases:
Potential Steps (reflection)
Potential Barriers (transmission and tunneling)
Potential Wells
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3.2 Particles & boxes: Potential Steps
Case 1: (partial reflection). Then:E > V0
k1
k2
=
=
2mE
ℏ2
− −−−−√2m(E − )V0
ℏ2
− −−−−−−−−−√(x) = +ψ1 A1e
ı xk1 A~
1e−ı xk1
(x) = +ψ2 A2eı xk2 A
~2e
−ı xk2
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3.2 Particles & boxes: Potential Steps
Transmission and reflection at potential steps [Case 1: . (cont'd)]
Now fix constants via boundary conditions. Since the Schrödinger equationis homogenous, we can only determine the three ratios , ,
.
We have three constants, but only two conditions: and. We shall thus only consider the case , which
amounts to an incident particle coming from .(why? we have seen in HW6 that a wave function carries a probability density flux in
direction)
E > V0
/A~
1 A1 /A2 A1
/A~
2 A1
(0) = (0)ψ1 ψ2
(0) = (0)ddxψ1
ddxψ2 = 0A
~2
x = −∞e+ı ⋅k⃗
x ⃗ +k⃗
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3.2 Particles & boxes: Potential Steps
Transmission and reflection at potential steps [Case 1: . (cont'd)]
The conditions and then give:
: Superposition of two waves with wave vector . One is associatedwith the incident wave, the other with the reflected wave.
: Transmitted wave. (Note also the case.)
E > V0
k1
k2
=
=
2mE/ℏ2− −−−−−−√2m(E − )/V0 ℏ2− −−−−−−−−−−−√
(x) = +ψ1 A1eı xk1 A
~1e
−ı xk1
(x) =ψ2 A2eı xk2
(0) = (0)ψ1 ψ2 (0) = (0)ψ′1 ψ′
2
= =A~
1
A1
−k1 k2
+k1 k2
A2
A1
2k1
+k1 k2
ψ1 ±k1
ψ2 → 0V0
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3.2 Particles & boxes: Potential Steps
Transmission and reflection at potential steps
One can then define transmission ( ) and reflection ( ) coefficients. (Theseare consistent with the ratio of probability flux on both sides):
Note that
Unlike for a classical particle, which just reduces its momentum whenpassing a potential step, in QM a particle has a non-zero probability ofturning back.
T R
J
R
T
=
=
= … = 1 −∣
∣∣A~
1
A1
∣
∣∣2
4k1k2
( +k1 k2)2
= … =k2
k1
∣∣∣A2
A1
∣∣∣2 4k1k2
( +k1 k2)2
T + R = 1
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3.2 Particles & boxes: Potential Steps
Case 2: (total reflection). Then:E ≤ V0
k1
ρ2
=
=
2mE
ℏ2
− −−−−√2m( − E)V0
ℏ2
− −−−−−−−−−√(x) = +ψ1 A1e
ı xk1 A~
1e−ı xk1
(x) = +ψ2 B2exρ2 B
~2e
− xρ2
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3.2 Particles & boxes: Potential Steps
Total reflection at potential steps [Case 2: . (cont'd)]
For the solution to be normalizable, we must have (prefactor of .Otherwise as )
The conditions and then give:
The reflection coefficients becomes:
As in classical mechanics, the particle is reflected. But it does have a non-zero, exponentially decaying probability of being found in the classicallyforbidden region at .
E ≤ V0
= 0B2 e xρ2
| (x) → ∞ψ2 |2 x → ∞
(0) = (0)ψ1 ψ2 (0) = (0)ψ′1 ψ′
2
= =A~
1
A1
− ık1 ρ2
+ ık1 ρ2
B2
A1
2k1
+ ık1 ρ2
R = = = 1∣
∣∣A~
1
A1
∣
∣∣2
∣∣∣
− ık1 ρ2
+ ık1 ρ2
∣∣∣2
x > 0
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3.2 Particles & boxes: Potential Steps
Note: There is only one total WF. It is:
k1
ρ2
=
=
2mE
ℏ2
− −−−−√2m( − E)V0
ℏ2
− −−−−−−−−−√(x) = +ψ1 A1e
ı xk1 A~
1e−ı xk1
(x) = +ψ2 B2exρ2 B
~2e
− xρ2
ψ(x) = { (x) for x < 0ψ1
(x) for x ≥ 0ψ2
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3.2 Particles & boxes: Potential Barriers
Case 1: (resonances). Then:E > V0
k1
k2
k3
=
=
=
2mE/ℏ2− −−−−−−√2m(E − )/V0 ℏ2− −−−−−−−−−−−√
k1
(x) = +ψ1 A1eı xk1 A
~1e
−ı xk1
(x) = +ψ2 A2eı xk2 A
~2e
−ı xk2
(x) = +ψ3 A3eı xk1 A
~3e
−ı xk1
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3.2 Particles & boxes: Potential Barriers
Resonance trapping in potential barriers
Let us choose, as before, (i.e., particle incident from )
The matching conditions at :
then give and in terms of , and the matching conditions at :
give and in terms of and (and thus ).
= 0A~
3 x = −∞
x = l
(l) = (l) (l) = (l)ψ2 ψ3d
dxψ2
ddx
ψ3
A2 A~
2 A3 x = 0
(0) = (0) (0) = (0)ψ1 ψ2d
dxψ1
ddx
ψ2
A1 A~
1 A2 A~
2 A3
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3.2 Particles & boxes: Potential Barriers
Resonance trapping in potential barriers
(After some calculation) we find:
and...
A1
A~
1
=
=
(cos( l) − ı sin( l))k2+k2
1 k22
2k1k2k2 eı lk1 A3
ı sin( l)−k2
2 k21
2k1k2k2 eı lk1 A3
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3.2 Particles & boxes: Potential Barriers
Resonance trapping in potential barriers
With and we can compute the reflection coefficient and thetransmission coefficient :
Inserting and this gives:
/A~
1 A1 /A3 A1 R
T
R
T
=
=
=∣
∣∣A~
1
A1
∣
∣∣2 ( − sin( lk2
1 k22)2 k2 )2
4 + ( − sin( lk21k
22 k2
1 k22)2 k2 )2
= = 1 − R∣
∣∣A~
3
A1
∣
∣∣2 4k2
1k22
4 + ( − sin( lk21k
22 k2
1 k22)2 k2 )2
k1 k2
T =4E(E − )V0
4E(E − ) + sinV0 V 20 (l )2m(E − )/V0 ℏ2− −−−−−−−−−−−√ 2
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3.2 Particles & boxes: Potential Barriers
Resonance trapping in potential barriers
Transmission coefficient varies periodically with barrier length :
Maximum: whenever ( ) (i.e., whenever is aninteger multiple of the half wavelength )
Minimum whenever
T =4E(E − )V0
4E(E − ) + sinV0 V 20 (l )2m(E − )/V0 ℏ2− −−−−−−−−−−−√ 2
l
T = 1 l = nπk2 n ∈ Z l
π/k2
T = (1 + )V 20
4E(E− )V0
−1
l = (n + )πk212
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3.2 Particles & boxes: Potential Barriers
What is happening here? At resonance, the partial waves reflected at and are in constructive interference. Standing waves in region II!
Wave packet analysis would show that near resonance, wave packetsspends a long time in region II (resonance scattering).
x = 0x = l ⇒
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3.2 Particles & boxes: Potential Barriers
Case 1: (tunneling). Then:E ≤ V0
k1
ρ2
k3
=
=
=
2mE/ℏ2− −−−−−−√−2m(E − )/V0 ℏ2− −−−−−−−−−−−−−√
k1
(x) = +ψ1 A1eı xk1 A
~1e
−ı xk1
(x) = +ψ2 A2exρ2 A
~2e
− xρ2
(x) = +ψ3 A3eı xk1 A
~3e
−ı xk1
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3.2 Particles & boxes: Potential Barriers
Tunneling through potential barriers
(same calc as before... (or replace by )):
If , then , and we get:
The transmission coefficient is not zero, but decays exponentially with .
k2 −iρ2
T = =∣
∣∣A~
3
A1
∣
∣∣2
4E( − E)V0
4E(E − ) + sinhV0 V 20 (l )2m( − E)/V0 ℏ2− −−−−−−−−−−−√ 2
z = l ≫ 1ρ2 sinh(z) = ( − ) ≈12 ez e−z 1
2 ez
T ≈16E( − E)V0
V 20
e−2 lρ2
l
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3.2 Particles & boxes: Potential Barriers
Tunneling through potential barriers
What does that mean in practice? E.g., consider an electron. With theelectron mass, we get as decay length :
[ —the electron volt, is a microscopic unit of energy; it is the energygained by an electron when accelerated over a potential difference of .Note: . The unit , the Ångström, is exactly
(or ]
If an electron with then hits an energy barrier with height and length , we get a (large!) transmission probability of
1/ρ2
= ≈1ρ2
ℏ
2m( − E)V0− −−−−−−−−−√
1.96Å
( − E)/eVV0− −−−−−−−−−√
eV1V
1eV ≈ 1.602 ⋅ J ≈ 96.49kJ/mol10−19 Åm10−10 0.1nm
E = 1eV = 2eVV0
1Å
T ≈ 0.78 = 78%.
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3.2 Particles & boxes: Potential Barriers
Tunneling through potential barriers
For a proton, with a times larger mass than an electron, the decaylength decreases to
If a proton with then hits the same energy barrier (with height and length ), its transmission coefficient is only
That is quite a difference... the power of exponential scaling.
For macroscopic objects—with much larger masses—tunneling is thuspractically impossible.
≈ 1840
= ≈ .1ρ2
ℏ
2m( − E)V0− −−−−−−−−−√
4.6 ⋅ Å10−2
( − E)/eVV0− −−−−−−−−−√
E = 1eV= 2eVV0 1Å
T ≈ 4 ⋅ .10−19
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3.2 Particles & boxes: Potential Wells
Bound states: Let (other case: handled before w/ )− ≤ E ≤ 0V0 ≤ 0V0
ρ
k
=
=
−2mE/ℏ2− −−−−−−−√2m(E + )/V0 ℏ2− −−−−−−−−−−−√
(x) = +ψ1 A1eρx A
~1e
−ρx
(x) = +ψ2 A2eıkx A
~2e
−ıkx
(x) = +ψ3 A3eρx A
~3e
−ρx
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3.2 Particles & boxes: Potential Wells
Bound states in potential wells
Since must be bounded in region I, we need . The matchingconditions at then give:
and the conditions at :
ψ = 0A~
1
x = −a/2
A2
A~
2
=
=
e(−ρ+ık)a/2 ρ + ık
2ıkA1
− e−(ρ+ık)a/2 ρ − ık
2ıkA1
x = +a/2
A3
A1
A~
3
A1
=
=
((ρ + ık − (ρ − ık )e−ρa
4ıkρ)2eıka )2e−ıka
sin(ka)+ρ2 k2
2kρ
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3.2 Particles & boxes: Potential Wells
Bound states in potential wells
On the other hand, we also need , since otherwise is unboundedon the right hand side.
This leads to condition:
Since and depend on , this means that, unlike in the previous cases,only discrete values of lead to feasible (normalizable) wave functions. Thebound state energy is quantized.
(in general: bound state WFs are normalizable and quantized, unbound state WFs are not
normalizable (and thus not by themselves capable of representing physical states) and have
continuous spectra)
= 0A3 ψ(x)
= ⇔ = ±( )ρ − ık
ρ + ık
2
e2ıka ρ − ık
ρ + ıkeıka
ρ k E
E
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3.2 Particles & boxes
Summary
Important classes of quantum states (scattering states, bound states) andquantum behavior (transmission, reflection, tunneling, and resonant trapping of wave
packets) already emerge in simple piecewise constant potentials. These aresusceptible to analytical treatment.
Quantization and / coefficients arise from the Schrödinger inconjunction with boundary conditions on the wave function. (continuity of
, continuity of (unless ), normalizability (if bound state))
Treating these phenomena is helpful for making sense of quantum behaviorin complex problems
R T
ψ(x) ψ(x)∇⃗ V ( ) = ∞x⃗
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Grand Topic #3
3. Simple Model Problems
Free particles & Wave packets (Plane wave solutions; Superposition principle; Gaussian
wave packets)
Numerical solution of 1-particle quantum problems (Dimensionless EOM; Real-space
grid discretization; Time evolution; Quantum phenomena (eigenstates, tunneling, Husimi-
repr.); Pseudo-spectral methods; Basis function decomposition/1)
Particle in a box & co (1D Case: Steps, barriers, wells; Transmission/reflection,
resonance, tunneling; 3D Case: Separation of variables)
Vibrations & Harmonic oscillator (H.-o. (1D,3D), Polyatomic case, Non-harmonic
oscillator (1D) & QM approximation methods (subspace projections, perturbation expansion,
variational method)), outlook: field quantization
Spherical potentials & Angular momenta (Separation of diatomic nuclear Schrödinger
equation; Rotational spectra; Ro-vibrational spectra )
Spin & Two-level systems (Single spin ½; Coupled spins; Combining orbital & spin-
degrees of freedom)
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3.2 Particles & boxes: 3D box
The three-dimensional box
Let us consider the simplest case in 3 spatial dimensions: The 3D boxpotential with infinite potential walls (box sizes: ):
(One can similarly consider 3D boxes with periodic boundary conditions HW)
a × b × c
W(x, y, z) = { 0 if x ∈ [0, a] ∧ y ∈ [0, b] ∧ z ∈ [0, c]+∞ otherwise
→
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3.2 Particles & boxes: 3D box
The three-dimensional box
The wave function is zero outside of the box (since there).
Within the box, the WF is obtained from the Schrödinger equation for 1particle in position representation:
where
Note that nothing couples the space dimensions , , .
|ψ⟩( )r ⃗ a × b × c W = ∞
( − + )ψ(x, y, z) = Eψ(x, y, z)Δℏ2
2mW(x, y, z)
=0 inside
Δ = = + + (Laplace operator).∇⃗ 2 ∂2
∂x2
∂2
∂y2
∂2
∂z2
x y z
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3.2 Particles & boxes: 3D box
The three-dimensional box—Separation Ansatz
Ansatz for wave function: Product of functions in individual space directions
An ansatz like this is called separation ansatz (because individual variablesare treated with separate uncoupeld functions.)
Let us compute the Laplacian for this:
ψ(x, y, z) = f(x)g(y)h(z)
Δψ
ψ(x, y, z)∂2
∂x2
ψ(x, y, z)∂2
∂y2
ψ(x, y, z)∂2
∂z2
=
=
=
f“(x)g(y)h(z)
f(x)g“(y)h(z)
f(x)g(y)h“(z)
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3.2 Particles & boxes: 3D box
The three-dimensional box—Separation Ansatz
Inserting this into the Schrödinger equation , and diving by
we get:
Or, for example,
⇒ Δψ(x, y, z) = f“(x)g(y)h(z) + f(x)g“(y)h(z) + f(x)g(y)h“(z)
( Δ − E)ψ = 0−ℏ2
2m
ψ
− E = 0−ℏ2
2mf“(x)g(y)h(z) + f(x)g“(y)h(z) + f(x)g(y)h“(z)
f(x)g(y)h(z)
f(x)g(y)h(z)
f(x)g(y)h(z)
⇔ − − − − E = 0ℏ2
2mf“(x)
f(x)ℏ2
2mg“(y)
g(y)ℏ2
2mh“(z)
h(z)
− = + + E =:ℏ2
2mf“(x)
f(x)ℏ2
2mg“(y)
g(y)ℏ2
2mh“(z)
h(z)Ex
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3.2 Particles & boxes: 3D box
The three-dimensional box—Separation Ansatz
Note that the lhs depends on , and rhs on and . Neverthess, theexpressions are supposed to be equal for all , , .
That can only work if both sides evaluate to constants which do not dependon at all!
The other two functions can be similarly arranged to give:
− = + + E =:ℏ2
2mf“(x)
f(x)ℏ2
2mg“(y)
g(y)ℏ2
2mh“(z)
h(z)Ex
x y z
x y z
x, y, z
− = − = − =ℏ2
2mf“(x)
f(x)Ex
ℏ2
2mg“(y)
g(y)Ey
ℏ2
2mh“(z)
h(z)Ez
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3.2 Particles & boxes: 3D box
The three-dimensional box—Separation Ansatz
So with this separation of variables, we can decompose the one 3DSchrödinger equation into 3 one-dimensional Schrödinger equations, for the
directions separately:
If , , fulfill those equations, then the total fulfills
i.e., the 3D Schrödinger equation for
.
x, y, z
− f“(x) = f(x) − g“(x) = g(x) − h“(z) = h(z)ℏ2
2mEx
ℏ2
2mEy
ℏ2
2mEz
f g h ψ(x, y, z) = f(x)g(y)h(z)
( ( + + ) − ( + + )) f(x)g(y)h(z) = 0−ℏ2
2m∂2x ∂2
y ∂2z Ex Ey Ez
( Δ − E)ψ(x, y, z) = 0−ℏ2
2m
E = + +Ex Ey Ez
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3.2 Particles & boxes: 3D box
The three-dimensional box—Separation Ansatz
We have already dealt with the 1D-equations. The boundary conditions areas before (e.g., ).
The wave functions and energies are:
where , , . Any combination ofthose quantum numbers yields a feasible solution for the 3D box.
f(0) = f(a) = 0
f(x)
g(y)
h(z)
=
=
=
sin( )2a
−−√ πxnx
a
sin( )2b
−−√ πyny
b
sin( )2c
−−√ πznz
c
=Ex
n2xh
2
8ma2
=Ey
n2yh
2
8mb2
=Ez
n2zh
2
8mc2
∈ {1, 2, 3, …}nx ∈ {1, 2, 3, …}ny ∈ {1, 2, 3, …}nz
, ,nx ny nz
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3.2 Particles & boxes: 3D box
The three-dimensional box—Separation Ansatz
Re-combining the solutions in the individual directions:
Note that the wave functions are automagically normalized:
ψ(x, y, z)
E
=
=
sin( ) sin( ) sin( )8abc
− −−−√ πxnx
a
πyny
b
πznz
c
( + + )h2
8mn2x
a2
n2y
b2
n2z
c2
∭ |ψ(x, y, z) dx dy dz = |f(x) dx ⋅ |g(y) dy |h(z) dz = 1|2 ∫ a
0|2 ∫ b
0|2 ∫ c
0|2
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3.2 Particles & boxes: 3D box
Degeneracy of states—Cubic box
Let us consider the case of a cube, where . Then the possibleenergy eigenvalues are:
in any combination of , , .
Unlike in the 1D case, we here can get degeneracy: Multiple non-equivalentwave functions with the same energy (e.g., and
and all have the same energy)
The number of different orthogonal states with the same energy is calleddegeneracy of an energy level.
a = b = c
E = ( + + ) = ( + + )h2
8mn2x
a2
n2y
b2
n2z
c2
h2
8ma2n2x n2
y n2z
∈ {1, 2, 3, …}nx ∈ {1, 2, 3, …}ny ∈ {1, 2, 3, …}nz
= 2, = 1, = 1nx ny nz
= 1, = 2, = 1nx ny nz = 1, = 1, = 2nx ny nz
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3.2 Particles & boxes: 3D box
Degeneracy of states—Cubic box
Sometimes people distinguish between systematic degeneracy andaccidental degeneracy
Systematic degeneracy is degeneracy resulting from a symmetry of theproblem. For example, the states and
can be switched to one another by exchanging the and axes. A transformation under which the Hamiltonian is invariant.
Accidental degeneracy arises when two energy levels are degenerate buthave qualitatively different wave functions not related by simple symmetries.For example, the states and happen to have the same energy, but they qualitatively different wavefunctions not obtained from each other via symmetry transforms.
= 2, = 1, = 1nx ny nz
= 1, = 2, = 1nx ny nz x
y
= 7, = 4, = 1nx ny nz = 8, = 1, = 1nx ny nz
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3.2 Particles & boxes: 3D box
Notes on the separation ansatz
Note 1: Separation ansatz for partial differential equation (PDE) only worksin case of certain potentials and boundary shapes, such that thecoordinates can be uncoupled. It is not possible to solve all PDEs like that.But in problems with high symmetry (e.g., spherical/elliptical potentials,rectangular wells, etc.) it is very useful.
Note 2: How do we know that we get all possible solutions of the differentialequations like this? And not just some solutions which happen to have theform ?
Normally we can't—some solutions cannot be written like this. But due tolinearity, all of the 3D solutions can be decomposed into a linearcombination of such „axis-aligned“ functions
ψ(x, y, z) = f(x)g(y)h(z)
⇒
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Grand Topic #3
3. Simple Model Problems
Free particles & Wave packets (Plane wave solutions; Superposition principle; Gaussian
wave packets)
Numerical solution of 1-particle quantum problems (Dimensionless EOM; Real-space
grid discretization; Time evolution; Quantum phenomena (eigenstates, tunneling, Husimi-
repr.); Pseudo-spectral methods; Basis function decomposition/1)
Particle in a box & co (1D Case: Steps, barriers, wells; Transmission/reflection,
resonance, tunneling; 3D Case: Separation of variables)
Vibrations & Harmonic oscillator (H.-o. (1D,3D), Polyatomic case, Non-harmonic
oscillator (1D) & QM approximation methods (subspace projections, perturbation expansion,
variational method)), outlook: field quantization
Spherical potentials & Angular momenta (Separation of diatomic nuclear Schrödinger
equation; Rotational spectra; Ro-vibrational spectra )
Spin & Two-level systems (Single spin ½; Coupled spins; Combining orbital & spin-
degrees of freedom)
Quantum Chemistry 1 (CHEM 565), Fall 2018
54 of 54