BCMB/CHEM 8190
Spin Operators and QM Applications
Quantum Description of NMR
Experiments
• Not all experiments can be described by
Bloch equations
q
– scalar coupling
p g examples
p
• Hamiltonians and Schrodinger’s equation
• Density matrix and Lioville
Lioville-VonNeuman
VonNeuman eq
• Product operators and transformation rules
• INEPT and HSQC examples
Some Pulse Sequences Have Classical (Bloch) Explanations
Spin-Echo Experiment – J Coupling Effects
180° pulse refocus chemical shifts, inhomogenieties, not Js
90x
180y
t1/2
z
t1/2
z
(observe, t2)
z
z
ω+J/2
x
y x
y
ω–J/2 ω+J/2
x
y
ω–J/2
x
This can b
Thi
be used
d tto measure T2 relaxation
l
ti or
scalar coupling in a 2D experiment
y
J t1
Some Sequences Need Quantum Explanations
Mixing
g by
y Scalar Coupling
p g of Directly
y Bonded Nuclei:
the INEPT and HSQC Experiments
1H
90x
180yy
90yy
15N
z
x
z
J->
y x
z
z
90x z
yx
y
z
J->
x
y x
yx
y
1H
15N
Quantum Mechanics Fundamentals (Spin Operations)
Expectation values correspond to observables:
E
μ = <ψ| μ |ψ> = ⌠ψ* μ ψ dτ
⌡
μ - an operator, ψ - a wave function (spin function)
β
α
B0
Examples:
p
ψ = α, β (one spin ½ ) ψ = αα, αβ, βα, ββ (two spins ½ )
solutions to Schrodinger’s equation: H ψ = E ψ
μ = γ I (h/2π)
μz = γ Iz (h/2π) (magnetic moments)
(in terms of spin operators)
Operations:
Iz (h/2π) |α> = (h/2π) ½ α, , Iz (h/2π) |β> = -(h/2π) ½ β
Hz = - μ•B0, Ez = < α|- μ•B0 |α> = -½ γ(h/2π) B0<α*|α>
= -½ γ(h/2π) B0
Hamiltonian Operator Containing Primary
Observables for High Resolution NMR
H = -γB0∑i (1-σi)IZi + ∑j>i 2π J Ii · Ij + ∑j>i 2π Ii · D · Ij
chemical shift scalar coupling dipolar coupling
(Zeeman-also angular momentum splitting in rotational spectra)
Ii = Iix + Iiy + Iiz
In “first
first order”
order spectra scalar coupling term can be
approximated as:
∑j>i 2π J Iiz · Ijz
Spin functions (ψ = α,
α β,
β αβ,
αβ βα …)) are solutions to
Schrodinger’s Equation, Hψ = Eψ, with only Zeeman term
Some other spin operators:
Ix |α> = ½ β
Iy |α> = ½ iβ
Iy1 |α β > = ½ iββ
Iy1Iy2 |α β > = ¼ βα
Ix | β > = ½ α
Iy | β > = -½
½ iα
Iy2 |α β > = -½ iαα
Iz1Iy2 |α β > = -¼ iαα
Note: α, β, are not eigenfunctions of Hamiltonians (H)
that contain these operators.
Solution: Sets such as αα, βα, αβ, αα are complete
orthonormal sets.
sets
Any spin function can be written in terms of these
ψ = c11 αα + c22 αβ
β + c33 βα
β + c44 ββ = ∑j cj φj
More Operators
• I2 |ψ> = I(I+1)ψ = ¾ ψ for I = ½
I2 = IxIx + IyIy + IzIz
I2 |α> = Ix ½ β + Iy ½ i β + Iz ½ α
=¼α+¼α+¼α = ¾α
• Raising
R i i and
dL
Lowering
i O
Operators:
t
Ix = (I+ + I-)/2,
Iy = (I+ - I-)/2i
I+ |β> = Ix |β> + i Iy |β> = ½ α + ½ α = α
I+ |α
|α> = 0, I- |α
|α> = β, I- |β
|β> = 0,
Using Operators: Energy Levels for
an AX Spin System in Solution
• Hamiltonian:
H = -ihνAIAz -ihνXIXz + hJAXIAzIXz
νi = γB0(1-σi)/(2π)
E
ββ
βα
αβ
αα
• H|αα> = -ihνA ½ αα -ihνX ½ αα + hJAX ½ ½ αα
<αα| H |αα> = -ihνA ½ -ihνX ½ + hJAX ¼
<αβ| H |αβ> = -ihνA ½ +ihνX ½ - hJAX ¼
etc.
One Quantum Transitions:
• ΔEαβ→ββ
β ββ = -h(-νA -νX)/2 + ¼ h JAX
+h(νA -νX)/2 + ¼ h JAX
= hνA + ½ h JAX
• ΔEαα→βα = -h(-νA +νX)/2 - ¼ h JAX
+h( A +ν
+h(ν
+ X)/2 - ¼ h JAX
= hνA - ½ h JAX
JAX
JAX
νA
νX
Operators in Matrix Notation
If we stay with one basis set, properties vary only because of
changes in the coefficients weighting each basis set function
μx = γ(h/2π)<ψ| Ix |ψ>
ψ= c1 αα + c2 αβ + c3 βα + c4 ββ = ∑j cj φj
<ψ|
ψ| Ix |ψ
|ψ> = ∑j,k
φj| Ix |φk>
j k cj* ck <φ
We need calculate <φj| Ix |φk> only once if we stay
with
ith this basis set – these can be pputt in a n x n matrix.
matri
Matrix equivalent: <ψ| Ix |ψ> = (c1, c2, …)* ⎡ ⎤ ⎛c1⎞
⎟ Ix⎟⎟ ⎟ c2⎟⎟
⎣ ⎦ ⎝ •⎠
Special Case: Pauli Spin Matrices
0
|Ix| =
½
½
0
0
|Iy| =
i½
-i½
0
Note: < α | Ix | α> = ½ < α | β > = 0
< α | Ix | β > = ½ < α | α > = ½
½
|Iz| =
0
0
½
How do they work? Try something we know: Ix | α > = ½ β
0
½
½
0
1
0
=
0
½
0
=½
1
= ½ β
Operators are a matrix of numbers, Spin functions a vector of numbers
Larger Collections of Spin ½ Nuclei
IAx =
αα
βα
αβ
ββ
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
|β =
<αα|| IAx|βα>
<αα| ½ |αα> = ½
αα βα αβ ββ
0 1/ 2 0
=
1/ 2 0
0 0
0
0
0 0
0 1/ 2
0 1/ 2 0
0
IXx =
0 1/ 2 0
0 0
1/ 2 0
0 1/ 2
0 0
0 1/ 2 0
0
Easier way: direct products: E ⊗ IAx with 2X2 matrices