BCMB/CHEM 8190
Spin Operators and QM Applications
Quantum Description of NMR
Experiments
• Not all experiments can be described by
Bloch equations – scalar coupling examples
• Hamiltonians and Schrodinger’s equation
• Density matrix and Lioville-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 be used to measure T2 relaxation or
scalar coupling in a 2D experiment
y
J t1
Some Sequences Need Quantum Explanations
Mixing by Scalar Coupling of Directly Bonded Nuclei:
the INEPT and HSQC Experiments
1H
τ
90x
τ = J/4
τ
180y
90y
15N
90x
z
x
z
J->
y x
z
z
z
yx
z
y
J->
x
y x
yx
y
Iz Æ -Iy Æ -2IxSz Æ -2IzSy
1H
15N
HSQC Spectrum of H-N Amides in a Protein
Red – without PI(4)P
Blue – with PI(4)P
Phosphoinositide interactions with PH domain of FAPP1 at a bicelle surface
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:
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
Ii = Iix + Iiy + Iiz
In “first order” spectra scalar coupling term can be
approximated as:
∑j>i 2π J Iiz · Ijz
Spin functions (ψ = α, β, 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 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.
Any spin function can be written in terms of these
ψ = c1 αα + c2 αβ + c3 βα + c4 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 and Lowering Operators:
Ix = (I+ + I-)/2,
Iy = (I+ - I-)/2i
I+ |β> = Ix |β> + i Iy |β> = ½ α + ½ α = α
I+ |α> = 0, I- |α> = β, I- |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 = -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(νA +νX)/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| Ix |ψ> = ∑j,k cj* ck <φj| Ix |φk>
We need calculate <φj| Ix |φk> only once if we stay
with this basis set – these can be put in a n x n matrix.
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
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
<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
IXx =
0
0
0 1/ 2 0
0 0 1/ 2
1/ 2 0 0
0 1/ 2 0
0
0
Easier way: direct products: E ⊗ IAx with 2X2 matrices