Zitterbewegung: diferència entre les revisions
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Línia 9:
==Teoria==
L'[[equació de Schrödinger]] depenent del temps
:<math> H \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi}{\partial t} (\mathbf{x},t) \,\!</math>
on <math> H \,\!</math> és el [[Hamiltonià (mecànica quàntica)|Hamiltonià]] de Dirac per a un electró en el espai lliure
:<math> H = \left(\alpha_0 mc^2 + \sum_{j = 1}^3 \alpha_j p_j \, c\right) \,\!</math>
implica que qualsevol operador Q obeeix l'equació
:<math> -i \hbar \frac{\partial Q}{\partial t} (t)= \left[ H , Q \right] \,\!\;.</math>
In particular, the time-dependence of the [[position operator]] is given by
:<math> \hbar \frac{\partial x_k}{\partial t} (t)= i\left[ H , x_k \right] = c\alpha_k \,\!\;</math>
where <math>\alpha_k \equiv \gamma_0 \gamma_k</math>.
The above equation shows that the operator <math>\alpha_k</math> can be interpreted as the kth component of a "velocity operator."
The time-dependence of the velocity operator is given by
:<math> \hbar \frac{\partial \alpha_k}{\partial t} (t)= i\left[ H , \alpha_k \right] = 2[i \gamma_k m - \sigma_{kl}p^l] = 2i[p_k-\alpha_kH] \,\!\;</math>
where <math>\sigma_{kl} \equiv \frac{i}{2}[\gamma_k,\gamma_l]</math>.
Now, because both <math>p_k</math> and <math>H</math> are time-independent, the above equation can easily be integrated twice to
find the explicit time-dependence of the position operator. First:
:<math>\alpha_k (t) = \alpha_k (0) e^{-2 i H t / \hbar} + c p_k H^{-1} </math>
Then:
:<math> x_k(t) = x_k(0) + c^2 p_k H^{-1} t + {1 \over 2 } i \hbar c H^{-1} ( \alpha_k (0) - c p_k H^{-1} ) ( e^{-2 i H t / \hbar } - 1 ) \,\!</math>
where <math> x_k(t) \,\!</math> is the position operator at time <math> t \,\!</math>.
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