Quantum-Mechanical Description

Whereas in classical mechanics all information about the state of a physical system is fixed by a point in its phase space, in quantum mechanics the state of a system is represented by a so-called state vector |ψi, which is defined in a complex vector space. This vector space is called Hilbert space H. Following the developments of P.A.M. Dirac [19], |ψi is referred to as aket vector andhψ| as abra vector. Both are defined in their own but dual Hilbert space and are defined as to contain all the information about the physical system.

Classical observables such as angular momentum J are defined to be represented by linear operators, like the angular momentum operator ˆJ. These operators are defined in the Hilbert space of the corresponding physical system which is defined by the state vector

|ψi.

In general an operator acting on a state vector is not keeping the state vector in its original form. However, there are particular kets of importance, known as eigenkets of an operator with the property

Aˆ|a1i = a1|a1i

Aˆ|a₂i = a₂|a₂i (2.8)

... ... ...

wherea₁, a₂,· · · are scalars and called eigenvalues of the operator ˆA. The physical state corresponding to an eigenket is called eigenstate. From this it is clear that the vector space over which an operator ˆA is defined is spanned by the N-dimensional basis of its eigenkets |aii. Further it is postulated that all observables are represented by hermitian operators ˆA

Aˆ≡Aˆ^† (2.9)

and hence have purely real eigenstates. Eq. (2.9) also implies that the set of eigenkets {|a_ii} forms an orthonormal basis of the Hilbert space and therefore obeys

hai|aji=δ_ij (2.10)

This is a reasonable postulate since we identified operators with the classical observables and therefore the value measured corresponds to the eigenvaluesai which in turn have to be real. Using this, any arbitrary ket|ψi can be linearly expanded as

|ψi = X

i

|aii hai| |ψi

= X

i

cai|aii (2.11)

in the basis of the eigenkets|aiiof the operator ˆA, wherecai are in general complex scalars.

Looking at Eq. (2.11)

Λˆai ≡ |aii hai| (2.12)

can be interpreted as a projection operator which, because of the completeness of the basis

|aii, fulfils

X

i

|aii hai|= _dim(^Aˆ) (2.13) Aˆis easiest represented as a matrix in the basis of its eigenstates.

Aˆ = X

i

X

j

|a_iiD a_i

Aˆ|a_jE ha_j|

Aˆ_ij = a_jha_i|a_jiδ_ij (2.14)

In general this it not the case and also off-diagonal elements of ˆA are non-zero.

The measurement of an observable ˆA of a system in state|ψi puts the system into the state|a_ii

Dψ Aˆ|ψE

= X

i

hψ|aiiD ai

Aˆ|aj

Ehaj|ψi

= a_i|ha_i|ψi|² (2.15)

yielding the eigenvalue ai. Every following measurement on the same system will now yield the same eigenvalue since |ψi (Eq. (2.11)) has been reduced to one eigenstate |aii of the operator ˆA (state reduction). The probability of measuring the eigenvalue a_i on a system in state |ψi is given in accordance with Eq. (2.15) by

|ha_i|ψi|² =|c_i|² (2.16) The time evolution of a state|ψ(t)i is given by the time-dependent Schr¨odinger equation

i~∂

∂t|ψ(t)i= ˆH |ψ(t)i (2.17) where the Hamilton operator ˆH( ˆp,q) is derived from the classical Hamilton functionˆ H(p,q) by replacing the canonical conjugate variables by operators (correspondence prin-ciple).

The equation of motion for a particle in a magnetic fieldB₀ with a potentialA is i~∂

∂t|ψ(t)i = 1

2mπˆ²|ψ(t)i (2.18)

ˆ

π = pˆ−eAˆ (2.19)

where ˆπ is a generalised canonical momentum. While this equation is readily capable of describing the orbital angular momentum ˆL, it does not yet involve the intrinsic spin ˆS of the nucleus. Motivated by Stern-Gerlach experiments which suggested the existence of operators that have two eigenstates, Pauli introduced the so-called Pauli spin matrices [20, 5]

ˆ

Eq. (2.22), the anticommutator, is special for spins S = ¹₂ (fermions) and Eq. (2.21) is defining an angular momentum algebra. Pauli replaced the classical generalised momen-tum ˆπ in Eq. (2.18) by ˆσπˆ using the Pauli spin matrices, yielding

Now the first term ˆH^π describes the classical generalised momentum ˆπ of the particle, which will be neglected from now on since any orbiting motion of the nucleus itself shall be neglected. The second term is describing an angular momentum and can be written as

Hˆ^S = −~e 2mσBˆ ₀

= −γSSBˆ ₀ (2.26)

whereγS is the quantum-mechanical gyromagnetic ratio. It is important to mention that the quantum mechanical gyromagnetic ratio is not given exactly by _2m^~e (see Section 2.1.1).

Eq. (2.26) gives the key to the correspondence principle to convert the classical magnetic moment to the quantum mechanical operator

µ→γS~Sˆ (2.27)

Sˆ will from now on be referred to as spin operator which fulfils, together with its corre-sponding eigenstates |S, m_Si, the following eigenequations

Sˆz|S, mSi = ~mS

where|S, mSi are the two common eigenstates of both the squared spin operator ˆS² and its z-component. ˆS² and ˆSz together form the complete set of compatible observables of a spin S = ¹₂. It follows that every possible orientation of spin ˆS must be representable by a linear superposition of the two eigenstates of ˆS² and ˆSz. The most general state of

a spinS= ¹₂, represented in the eigenbasis defined by Eq. (2.28), is where the phase factorsc_±¹

2 are related as

and α and β are the azimuth and altitude of the spin orientation. Thus, Eq. (2.32) describes what is called a coherent superposition (coherence) of the eigenstates

¹

is describing a spin pointing in the positivex-direction.

The solution to the Schr¨odinger equation of a single spin in a magnetic field is i~∂

∂t|ψi = −γSˆzB₀|ψi (2.34) with

|ψi= e^~iγSˆzB0t|S, m_Si= e^{−~iω0Sˆzt}|S, m_Si (2.35) whereB₀ =B₀ez and ω₀ =−γB₀ (compare Eq. (2.6)).

This is as far as one can go with a single spin. When being concerned with more than one spin all degrees of freedom (eigenstates) of every spin need to be preserved and the common Hilbert space is constructed by combining the Hilbert spaces of the single spins by a tensorial product

H=H^S1⊕H^S2 ⊕. . . (2.36) where the dimension of the new Hilbert space is (2S₁+ 1)·(2S₂+ 1)·. . . and the Hamil-tonian for two uncoupled spins reads as

Hˆ^S1S2 = ω₀^S1Sˆ_1z⊕ 2+ ₂⊕ω₀^S2Sˆ_2z (2.37) When combining the two Hilbert spaces of the spin operators ˆS₁ and ˆS₂

Sˆ = Sˆ₁⊕ ²+ 2⊕Sˆ₂ (2.38)

there exist two sets of mutually compatible observables and their respective eigenstates Sˆ₁²|S1S2;mS1mS2i = ~²S1(S1+ 1)|S1S2;mS1mS2i (2.39) Sˆ_1z|S₁S₂;m_S1m_S2i = ~m₁|S₁S₂;m_S1m_S2i (2.40) Sˆ₂²|S₁S₂;m_S1m_S2i = ~²S₂(S₂+ 1)|S₁S₂;m_S1m_S2i (2.41) Sˆ_2z|S₁S₂;m_S1m_S2i = ~m₂|S₁S₂;m_S1m_S2i (2.42)

and Finally it is useful to make the distinction between cases where spins are indistinguish-able, the so-called homonuclear case

hSˆ_1i,Sˆ_2ji

=i~_ijkSˆ_k; (2.51)

and the case of distinguishable spins, the so-called heteronuclear case hSˆ_1i,Sˆ_2ji

= 0 (2.52)

Up to this point no interactions between spins have been considered and the Hamiltonian Hˆ^S contains no structural information at all. In the following Section a closer look at the nuclear spin interactions will be taken.

Im Dokument MAS NMR of Nuclei with Spin S=1/2 in Polycrystalline Powders: Experiments and Numerical Simulations (Seite 16-20)