1 Up Next: Periodic Table Molecular Bonding PH300 Modern Physics SP11 “Science is imagination constrained by reality.” - Richard Feynman Day 24,4/19: Questions? H-atom and Quantum Chemistry Final Essay There will be an essay portion on the exam, but you don’t need to answer those questions if you submit a final essay by the day of the final: Sat. 5/7 Those who turn in a paper will consequently have more time to answer the MC probs. I will read rough draft papers submitted by class on Tuesday, 5/3 3 Recently: 1. Quantum tunneling 2. Alpha-Decay, radioactivity 3. Scanning tunneling microscopes Today: 1. STM’s (quick review) 2. Schrödinger equation in 3-D 3. Hydrogen atom Coming Up: 1. Periodic table of elements 2. Bonding energy SAMPLE METAL Tip SAMPLE (metallic) tip x Look at current from sample to tip to measure distance of gap. - Electrons have an equal likelihood of tunneling to the left as tunneling to the right -> no net current sample - Correct picture of STM-- voltage applied between tip and sample. energy I SAMPLE METAL Tip V I + sample tip applied voltage SAMPLE (metallic) sample tip applied voltage I SAMPLE METAL Tip V I + What happens to the potential energy curve if we decrease the distance between tip and sample?
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Up Next: Periodic Table
Molecular Bonding
PH300 Modern Physics SP11
“Science is imagination constrained by reality.”!- Richard Feynman!
Day 24,4/19: Questions? H-atom and Quantum Chemistry
Final Essay
There will be an essay portion on the exam, but you don’t need to answer those questions if you submit a final essay by the day of the final: Sat. 5/7
Those who turn in a paper will consequently have more time to answer the MC probs.
I will read rough draft papers submitted by class on Tuesday, 5/3
How is it same or different than Bohr, deBroglie? (energy levels, angular momentum, interpretation)
What do wave functions look like? What does that mean?
Extend to multi-electron atoms, atoms and bonding, transitions between states.
How does
Relate to atoms?
−2
2m∂2Ψ x,t( )
∂x2+V x,t( )Ψ x,t( ) = i ∂Ψ x,t( )
∂t
Apply Schrodinger Equation to atoms and make sense of chemistry!
(Reactivity/bonding of atoms and Spectroscopy)
How atoms bond, react, form solids? Depends on:
the shapes of the electron wave functions the energies of the electrons in these wave functions, and how these wave functions interact as atoms come together.
Next:
Schrodinger predicts: discrete energies and wave functions for electrons in atoms
5
What is the Schrödinger Model of Hydrogen Atom?
Electron is described by a wave function Ψ(x,t) that is the solution to the Schrodinger equation:
),,,(),,,(),,(
),,,(2 2
2
2
2
2
22
tzyxt
itzyxzyxV
tzyxzyxm
Ψ∂∂=Ψ+
Ψ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂+
∂∂+
∂∂−
2/1222
22
)(),,(
zyxZke
rZkezyxV
++−=−=
where: V r
Can get rid of time dependence and simplify: Equation in 3D, looking for Ψ(x,y,z,t):
Since V not function of time: /),,(),,,( iEtezyxtzyx −=Ψ ψ
/),,( iEtezyxE −ψ
),,,(),,,(),,(
),,,(2 2
2
2
2
2
22
tzyxt
itzyxzyxV
tzyxzyxm
Ψ∂∂=Ψ+
Ψ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂+
∂∂+
∂∂−
),,(),,(),,(),,(2 2
2
2
2
2
22
zyxEzyxzyxVzyxzyxm
ψψψ =+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂+
∂∂+
∂∂−
Time-Independent Schrodinger Equation:
Quick note on vector derivatives Laplacian in cartesian coordinates:
Laplacian in spherical coordinates:
Same thing! Just different coordinates.
3D Schrödinger with Laplacian (coordinate free):
ψψφψ
θθψθ
θθ
ψ
ErVmr
rr
rrm
=+⎥⎦
⎤⎢⎣
⎡∂∂+⎟
⎠⎞⎜
⎝⎛
∂∂
∂∂−
⎟⎠⎞⎜
⎝⎛
∂∂
∂∂−
)(sin1sin
sin1
2
12
2
2
22
2
22
2
Since potential spherically symmetric , easier to solve w/ spherical coords:
x
y
z
θ
φ
r
(x,y,z) = (rsinθcosϕ, rsinθsinϕ, rcosθ)
)()()( ),,( φθφθψ gfrRr =
Schrodinger’s Equation in Spherical Coordinates & w/no time:
Technique for solving = Separation of Variables
/iEte−)()()(),,,( φθφθ gfrRtr =Ψ
V (r) = −Zke2 / r( ) Note: physicists and engineers may use opposite definitions of θ and ϕ… Sorry!
In 3D, now have 3 degrees of freedom: Boundary conditions in terms of r,θ,φ
x y
z
θ
φ
r
What are the boundary conditions on the function R(r) ? a. R must go to 0 at r=0 b. R must go to 0 at r=infinity c. R at infinity must equal R at 0 d. (a) and (b)
ψ must be normalizable, so needs to go to zero … Also physically makes sense … not probable to find electron there
ψ (r,θ,φ) = R(r) f (θ)g(φ)
In 3D, now have 3 degrees of freedom: Boundary conditions in terms of r,θ,φ x
y
z
θ
φ
r
What are the boundary conditions on the function g(φ)? a. g must go to 0 at φ =0 b. g must go to 0 at φ=infinity c. g at φ=2π must equal g at φ=0 d. A and B e. A and C
How many quantum numbers are there in 3D? In other words, how many numbers do you need to specify unique wave function? And why? a. 1 b. 2 c. 3 d. 4 e. 5
Answer: 3 – Need one quantum number for each dimension:
(If you said 4 because you were thinking about spin, that’s OK too. We’ll get to that later.)
r: n θ: l ϕ: m
In 1D (electron in a wire): Have 1 quantum number (n)
In 3D, now have 3 degrees of freedom: Boundary conditions in terms of r,θ,φ
In 1D (electron in a wire): Have 1 quantum number (n)
In 3D, now have 3 degrees of freedom: Boundary conditions in terms of r,θ,φ Have 3 quantum numbers (n, l, m)
)()()(),,( φθφθψ mlmnlnlm gfrRr =x
y
z
θ
φ
r
In 1D (electron in a wire): Have 1 quantum number (n)
In 3D, now have 3 degrees of freedom: Boundary conditions in terms of r,θ,φ Have 3 quantum numbers (n, l, m)
ψ nlm (r,θ,ϕ ) = Rnl (r)Ylm θ,φ( )x
y
z
θ
φ
r
“Spherical Harmonics”
Solutions for θ & ϕ dependence of S.E. whenever V = V(r) è All “central force problems”
In 1D (electron in a wire): Have 1 quantum number (n)
In 3D, now have 3 degrees of freedom: Boundary conditions in terms of r,θ,φ Have 3 quantum numbers (n, l, m)
ψ nlm (r,θ,ϕ ) = Rnl (r)Ylm θ,φ( )x
y
z
θ
φ
r
Shape of ψ depends on n, l ,m. Each (nlm) gives unique ψ
2p
n=2 l=1
m=-1,0,1
n=1, 2, 3 … = Principle Quantum Number
l=0, 1, 2, 3 …= Angular Momentum Quantum Number =s, p, d, f (restricted to 0, 1, 2 … n-1) m = ... -1, 0, 1.. = z-component of Angular Momentum (restricted to –l to l)
Comparing H atom & Infinite Square Well: Infinite Square Well: (1D) • V(x) = 0 if 0<x<L
∞ otherwise
• Energy eigenstates:
• Wave functions:
H Atom: (3D) • V(r) = -Zke2/r
• Energy eigenstates:
• Wave functions:
2
222
2mLnEnπ=
Ψ n (x,t) =ψ n (x)e− iEnt /
)sin()( 2Lxn
Ln xπψ = ψ nlm (r,θ,φ) = Rnl (r)Ylm (θ,φ)
/),,(),,,( tiEnlmnlm
nertr −=Ψ φθψφθ
r
0 L
∞ ∞
x
22
422
2 nekmZEn
−=
7
What do the wave functions look like? ψ nlm (r,θ,φ) = Rnl (r)Ylm (θ,φ)l (restricted to 0, 1, 2 … n-1)
m (restricted to –l to l)
n = 1, 2, 3, …
n=1
s (l=0) p (l=1) d (l=2)
See simulation: falstad.com/qmatom
m = -l .. +l changes angular distribution
Much harder to draw in 3D than 1D. Indicate amplitude of ψ with brightness.
n=2
n=3
Increasing n: more nodes in radial direction
Increasing l: less nodes in radial direction; More nodes in azimuthal direction
Shapes of hydrogen wave functions:
ψ nlm (r,θ,φ) = Rnl (r)Ylm (θ,φ)Look at s-orbitals (l=0): no angular dependence
n=1 n=2
n=1 l=0
n=2 l=0
n=3 l=0
Higher n à average r bigger à more spherical shells stacked within each other à more nodes as function of r
Radius (units of Bohr radius, a0)
0.05nm
Probability finding electron as a function of r
P(r)
a) Zero b) aB c) Somewhere else
ψ nlm (r,θ,φ) = Rnl (r)Ylm (θ,φ)
probable
An electron is in the ground state of hydrogen (1s, or n=1, l=0, m=0, so that the radial wave function given by the Schrodinger equation is as above. According to this, the most likely radius for where we might find the electron is:
V = dV = dr( ) ⋅∫∫ r dθ( ) r sinθ dφ( ) = 4πr 2 dr∫
Ψ
2dV = ρ[r,θ ,φ] dV → P[r0 ≤ r ≤ r0 + dr] = 4πr0
2dr ⋅ R(r0 )2
d) 4πr2 dr
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In the 1s state, the most likely single place to find the electron is:
A) r = 0 B) r = aB C) Why are you confusing us so much?