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Basic Spin Quantum Mechanics

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NMR Spectroscopy and Imaging Department of Physics, FU Berlin Lecture 5: Basic Spin Quantum Mechanics 1 Leibniz-Institut für Molekulare Pharmakologie Lecture 5 1. Some Math Tools 2. QM Concepts 3. Nuclear Spin Hamiltonian 4. Uncoupled Spin-½ Nuclei 2 Leibniz-Institut für Molekulare Pharmakologie 5.1 Some Math Tools function properties, Dirac notation functions important to spin quantum mechanics are normalized and orthogonal which is described for a whole set of (eigen)functions by short Dirac notation for dealing later with state products etc. orthonormality adjoint states 3 Leibniz-Institut für Molekulare Pharmakologie 5.1 Some Math Tools operators, eigenstates matrix representation of operators an operator returns the same function this function is an eigenfunction trace (requires finite basis) the commutator is defined to describe the effect of changing the order of two consecutive operators the complex exponential of an operator is defined as 4 Leibniz-Institut für Molekulare Pharmakologie 5.1 Some Math Tools adjoint operators two operators are called adjoint if their matrix elements are related through and therefore also an operator that is self-adjoint is called Hermitian if the adjoint is the inverse of the same operator, it is called unitary 5 Leibniz-Institut für Molekulare Pharmakologie 5.1 Some Math Tools vector representation of functions suppose a function is given as linear combination of other functions this is written as with the coefficients 6 Leibniz-Institut für Molekulare Pharmakologie 5.1 Some Math Tools eigenbasis of commuting operators suppose we have eigenvectors and all the eigenvalues are non-degenerate and for another operator we know then the eigenbasis of A is also an eigenbasis for B or: A and B are both diagonalized using the eigenbasis of A 7 Leibniz-Institut für Molekulare Pharmakologie 5.2 QM Concepts the eigenstate basis a system is not necessarily IN one of the eigenstates but in any superposition (provided the eigenstates are normalized) however, the eigenstates play a special role when dealing with time-independent Hamiltonian the solution is and if the system was at the beginning in an eigenstate of the Hamiltonian, i. e. it will also be an eigenstate at time t, in other words: it is in a stationary state 8 Leibniz-Institut für Molekulare Pharmakologie 5.2 QM Concepts angular momentum define the angular momentum operators as with etc. cummutation properties and eigenstates characterized by the quantum numbers 9 Leibniz-Institut für Molekulare Pharmakologie 5.2 QM Concepts angular momentum an operator commuting with lz is l² with 10 Leibniz-Institut für Molekulare Pharmakologie 5.2 QM Concepts shift operators define two additional operators these affect the eigenstates as follows then we also get 11 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian spin hypothesis working with the full version of the wave function describing a system, one realizes that this is quite useless transition to a simplified version this includes only the terms depending on nuclear spin states and works with an averaged picture of electric and magnetic terms of the electrons this simplification, the spin Hamiltonian hypothesis, is justified because the electrons move fast on the time scale of nuclear motions and because the nuclear spin energies are only small compared to electron spin energies the Hamiltonian splits into electric and magnetic interactions 12 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian magnetic field interactions suppose we apply an external magnetic field of inductivity B then the interaction energy of a magnetic moment with this field is and the relation between the magnetic moment and the angular momentum operator is 13 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian interaction magnitudes the interactions governing the spin Hamiltonian can be of internal or external nature for spin ½, we only deal with magnetic interactions it should be emphasized that under most conditions the coupling of the spin system to the external field is stronger than the internal interactions this is the opposite situation to other spectroscopy techniques where the system is probed by weak perturbations and governed by internal interactions 14 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian types of external magnetic fields the total external field contains contributions from the static magnetic field, the RF field to excite the spin system and possibly a gradient field that allows further encoding the dominant operator will be the one describing the static field let us assume then only the z component operator from the magnetic moment remains 15 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian spin operator – a closer look particular form of angular momentum with eigenstates for the z- component operator and shift operators note: the Zeeman eigenstates can be used to represent any state of the system but contrary to ‘ordinary’ angular momentum there is no visualization possible like the spherical harmonics illustrating the electron orbitals however, one can easily manipulate these states in real physics experiments 16 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian Zeeman eigenstates spins I represented by a 2I +1 finite basis, i.e. for I = ½ with the following behavior representation of the z-operator in matrix form is straight forward for the x- and y-operator, we have a closer look at the shift operators, where we had for their most general form for any angular momentum 17 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian Zeeman eigenstates then we get for the raising operator and the other matrix entries vanish because of the orthogonality and similar for the other operator hence, we get for the transverse operators in the basis of the Zeeman eigenstates 18 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian RF excitation operators in its most general form, the RF field is applied off axis from B0 Q: why do we need RF > 0? this field acts only for some precisely defined pulse duration 19 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian RF excitation operators remember decomposing the oscillating transverse component then we get three components for the transverse 20 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian RF excitation operators remember decomposing the oscillating transverse component then we get three components and the longitudinal 21 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian RF excitation operators the longitudinal and off-resonant-components can be safely neglected, so we get the approximation with the nutation frequency the RF coil works most efficient for RF = 90° since this is a time-dependent operator, we will come back to it later 22 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins Zeeman energy levels the dominating energy term is given by the external field experienced by the nucleus, i.e. including the correction for the chemical shift the energy levels of the Zeeman eigenstates then read the energy difference is described by the Larmor frequency 23 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins superposition states the nucleus is NOT restricted to the two eigenstates; it can be in any superposition with the normalized coefficients or in vector notation with the eigenvectors note: the bra state is a row vector illustrating the normalization 24 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins some special states for the following superpostion we try to measure the x component here, the x component is sharp whereas y and z would show random results 25 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins some special states and similar for where we try measure the y component this time, the y component is sharp whereas x and z show random results if B is along z, |x> and |-y> do not have sharp energies and cannot be represented in energy level diagrams 26 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins geometric representations although we can use the energy levels only for the eigenstates of the operator belonging to the quantization axis, a geometric illustration is possible for these special superpositions again: |> and |> are still orthonormal! 27 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins arbitrary superposition also the superposition follows an eigenequation: 28 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins phase factors suppose a state is given by multiplying another with just a phase factor like is related to Q: what is the phase factor? states related through a phase factor obey the same eigenequation note: the phase cannot be depicted in arrow diagrams -|> is |> with different phase but not |>! 29 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins state evolution we know that we measure a precession, so we expect a time-dependent spin state that obeys the Schrödinger equation assuming only free evolution, i.e. no rf field this first order differential equation is solved by with the exponential operator representing a rotation note: the Hamiltonian is time-independent in this case 30 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins rotation operators exponentials of the spin operators are called rotation operators – but what do they do? the complex exponential of an operator is defined as suppose we have three operators
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