Classical behavior of magnetic dipole vector. P. J. Grandinetti

Transcription

1 Classical behavior of magnetic dipole vector Z μ Y X Z μ Y X

2 Quantum behavior of magnetic dipole vector Random sample of spin 1/2 nuclei measure μ z μ z = + γ h/2 group μ z = γ h/2 group

3 Quantum behavior of magnetic dipole vector Randomly oriented spin 1 nuclei measure μ z μ z = + γ h group μ z = 0 group μ z = γ h group 33.3% 33.3% 33.3%

4 Quantum behavior of magnetic dipole vector Quantized observables is not in conflict with our picture of a spinning magnetic dipole vector undergoing continuous precession about an externally applied magnetic field. 0% μ z = + γ h/2 group 50% μ z = + γ h/2 group μ z = + γ h/2 group (π) x measure μ z μ z = + γ h/2 group (π/2) x measure μ z 100% μ z = γ h/2 group 50% μ z = γ h/2 group 100% μ y = + γ h/2 group μ z = + γ h/2 group (π/2) x measure μ y 0% μ y = γ h/2 group

5 Quantum Math Ket Space Bra Space { { dimension associated with μ z = - γ h /2 state vector dimension associated with μ z = - γ h /2 { { state vector dimension associated with μ z = + γ h /2 dimension associated with μ z = + γ h /2

6 Quantum Math

7 Quantum Math Similarly, using one obtains...

8 Quantum Math Assignment Commutator

9 Quantum Math When an operator acts on a state, the state can be transformed into another state or linear combination of states in the complex space. Eigenstates of an Operator There is a special set of states, know as eigenstates of an operator, which are not transformed into other states when that operator acts on them. When an operator acts on its eigenstate the same state remains multiplied by a real number Expectation Value of an Operator

10 Expectation values : Spin 1/2 Z μ φ <μ y> θ <μ z> <μ x> Y X

11 Quantum Dynamics Schrodinger Equation Time Propagator Hamiltonian Solve Schrodinger equation for time independent Hamiltonian: What is the Hamiltonian operator?

12 The Zeeman Hamiltonian How do we know our Hamiltonian? We start with the energy of a classical magnetic dipole subjected to a magnetic field and replace all observables by the corresponding operators If we take B as along the z axis we obtain

13 Solve the Schrodinger Equation: free precession Solve the Schrodinger equation for the free evolution of the nuclear magnetic dipole vector in the lab frame and with

14 Solve the Schrodinger Equation: free precession Solve the Schrodinger equation for the free evolution of the nuclear magnetic dipole vector in the lab frame

15 Projection Operator Two Level (outcome) system Calculate the expecation value of an operator

16 Projection Operator Two Level (outcome) system Calculate the expecation value of an operator

17 Projection Operator Two Level (outcome) system Calculate the expecation value of an operator Take the trace of a matrix

18 Trace of a Matrix

19 Projection Operator Two Level (outcome) system Calculate the expecation value of an operator Prove that trace of product of two matrices gives

20 Projection Operator Matrix representation of Observable operator What is this matrix?

21 Projection Operator Matrix representation of Observable operator What is this matrix? It is the matrix representation of the Projection operator, formed by taking the outer product of the state vector

22 Projection Operator Matrix representation of Observable operator What is this matrix? It is the matrix representation of the Projection operator, formed by taking the outer product of the state vector Expectation value for an operator is trace of product the operator with the projection operator

23 Projection Operator Dynamics Two Level System (Spin 1/2)

24 Meet the Density Operator What is the total I x expectation value for two identical non-interacting spins? Using Inner Products Using Projection Operator

25 Meet the Density Operator What is the total I x expectation value for N identical non-interacting spins? Density Operator

26 What is the Equilibrium Density Operator? Equilibrium Density Operator For NMR we have In the high temperature approximation

27 Rotations

28 One Pulse and Acquire (Bloch Decay) : Spin 1/2 Like Bloch equations, we transform density operator and Hamiltonian into the rotating frame to simplify our equations. During the Pulse if we assume then we calculate then density operator in rotating frame Setting and we have

29 One Pulse and Acquire (Bloch Decay) : Spin 1/2 After the Pulse more convenient to use Then we calculate the signal using add effect of T2 relaxation and we have: after FT:

30 Two Pulses and Acquire (Spin Echo) : Spin 1/2 when rf pulse is on, The initial density operator will be when rf pulse is off. and immediately after a π/2 pulse we have S(t) π/2 π After t 1 evolution we obtain time t 1 t 1 = t 2 t 2 Immediately after the π pulse we have and after t 2 evolution we obtain Calculating the signal from the trace we obtain when we have

31 One Pulse and Acquire : Two Coupled Spin 1/2 During the pulse we assume that for a period and obtain and apply After the pulse we have free evolution under which in the end leads to Take the trace of with and obtain

32 One Pulse and Acquire : Two Coupled Spin 1/2 which can be rewritten as 2 π J 2 π J Ω 1 Ω 2 ν

33 Two Pulses and Acquire : Two Coupled Spin 1/2 S(t) π/2 π 2 π J 2 π J time t 1 t 2 t 1 = t 2 Ω1 Ω2 2 π J 2 π J ν 1 Ω2 Ω1 ν 2

34 COSY: Two Coupled Spin 1/2 (COrrelation SpectroscopY - COSY) π/2 π/2 S(t) 2 π J 2 π J time t 1 t 2 t 1 = t 2 Ω1 Ω2 2 π J 2 π J ν 1 Ω2 Ω1 ν 2

35 More Rotations (J evolution) : Two coupled spins