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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Magnetic resonance (quantum mechanics) ： ウィキペディア英語版 Magnetic resonance (quantum mechanics) Magnetic resonance is a phenomenon that affects a Magnetic dipole when placed in a uniform static magnetic field. Its energy is split into a finite number of energy levels, depending on the value of quantum number of angular momentum. This is similar to energy quantization for atoms, say in H atom; in this case the atom, in interaction to an external electric field, transitions between different energy levels by absorbing or emitting photons. Similarly if a magnetic dipole is perturbed with electromagnetic field of proper frequency($E/\backslash ,$), it can transit between its energy eigenstates, but as the separation between energy eigenvalues is small, the frequency of the photon will be the microwave or radio frequency range. If the dipole is tickled with a field of another frequency, it is unlikely to transition. This phenomenon is similar to that, when a system is acted on by a periodic force of frequency equal to its natural frequency. ==Quantum mechanical explanation== As a magnetic dipole, using a spin $\backslash tfrac$ system such as a proton; according to the quantum mechanical state of the system, denoted by :$|\backslash Psi(t)\backslash rangle$ , evolved by the action of an unitary operator $e^$; the result obeys Schrödinger equation: $i\; \backslash hbar\; \backslash frac\backslash Psi\; =\; \backslash hat\; H\; \backslash Psi$ States with definite energy evolve in time with phase $e^$ ,( $|\backslash Psi(t)\backslash rangle=\; |\backslash Psi(0)\backslash rangle\; e^$ ) where E is the energy of the state, since the probability of finding the system in state $|\; \backslash langle\; x|\backslash Psi(t)\backslash rangle|^2$= $|\; \backslash langle\; x|\backslash Psi(0)\backslash rangle|^2$ is independent of time. Such states are termed stationary states, so if a system is prepared in a stationary state, (i.e. one of the eigenstates of the Hamiltonian operator), then P(t)=1,i.e. it remains in that state indefinitely. This the case only for isolated systems. When a system in a stationary state is perturbed, its state changes, so it is no longer an eigenstate of the system's complete Hamiltonian. This same phenomenon happens in magnetic resonance for a spin $\backslash tfrac$ system in a magnetic field. The Hamiltonian for a magnetic dipole $\backslash bold$ (associated with a spin $\backslash tfrac$ particle) in a magnetic field $\backslash bold\backslash hat$ is: Here $\backslash omega\_0\; :=\; \backslash gamma\; B\_0$ is the larmor precession frequency of the dipole for $\backslash bold$ magnetic field and $\backslash sigma\_z$ is z Pauli matrix. So the eigenvalues of $\backslash hat$ are $-\backslash tfrac\backslash omega\_0$ and $\backslash tfrac\backslash omega\_0$. If the system is perturbed by a weak magnetic field $\backslash bold$, rotating counterclockwise in x-y plane (normal to $\backslash bold$) with angular frequency $\backslash omega$, so that $\backslash bold=\backslash hatB\_1\; \backslash cos-\backslash hatB\_1\; \backslash sin$ , then $\backslash begin1\backslash \backslash 0\; \backslash end$ and $\backslash begin0\backslash \backslash 1\; \backslash end$ are not eigenstates of the Hamiltonian, which is modified into It is inconvenient to deal with a time-dependent hamiltonian. To make $\backslash hat$ time-independent requires a new reference frame rotating with $\backslash bold$ ,i.e. rotation operator $\backslash hat(t)$ on $|\backslash Psi(t)\backslash rangle$, which amounts to basis change in hilbert space. Using this on Schrödinger's equation, the Hamiltonian becomes: $\backslash hat=R(t)\backslash hat\; R(t)^\backslash dagger\; +\backslash tfrac\backslash omega\backslash sigma\_z$ Writing $\backslash hat$ in the basis of $\backslash sigma\_z$ as- Using this form of the Hamiltonian a new basis is found: where $\backslash Delta\backslash omega=\backslash omega-\backslash omega\_\; 0$ and $\backslash omega\_1=\backslash gamma\; B\_1$ This Hamiltonian is exactly similar to that of a two state system with unperturbed energies $\backslash tfrac\backslash Delta\backslash omega$ & $-\backslash tfrac\backslash Delta\backslash omega$ with a perturbation expressed by ; According to Rabi oscillation, starting with $\backslash begin1\backslash \backslash 0\; \backslash end$ state, a dipole in parallel to $\backslash bold$ with energy $-\backslash tfrac\backslash omega\_0$, the probability that it will transit to $\backslash begin0\backslash \backslash 1\backslash end.$ state (i.e. it will flip) is Now if $\backslash omega\; =\; \backslash omega\_0$, i.e. $\backslash bold$ rotates with larmor frequency of the dipole in $\backslash bold$ magnetic field, then at some point of time, say $t\; =\; \backslash frac0\backslash \backslash 1\; \backslash end$; i.e. a diplole can be completely flipped. When $\backslash omega\; \backslash not=\backslash omega\_0$, the probability of change of energy state is small, so the previous case reflected resonance. This resonance condition is used for measurement of the magnetic moment of a dipole, magnetic field at a point in space, etc. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Magnetic resonance (quantum mechanics)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.