The semiclassical treatment of an simple two level atom in a resonant electromagnetic field is usually done in the Schrodinger/Interaction picture of QM, by assuming that the wavefunction of the atom at time 't' is given as: \begin{equation} |\alpha\rangle= c_1(t)|g\rangle + c_2(t)|e\rangle \end{equation} where $|\alpha\rangle$ is the wavefunction of the system and $c_1(t)$ and $c_2(t)$ are just coefficients of the ground and excited state kets (with some other exponential factor also included if required, but that is not really essential to the question). The Hamiltonian is of the form: \begin{equation} H=H_0+ AF(t) \end{equation} where $A$ is a constant operator, and $F(t)$ is just an ordinary function of time, usually a sinusoidal function. Now, the Schrodinger equation of time evolution of kets is applied and differential equations are obtained for $c_1(t)$ and $c_2(t)$, which are solved to calculate various transition probabilities, etc.

Now, if we do the calculation in the Heisenberg picture instead of the Schrodinger picture, then the kets will not evolve with time, only the operators. Hence, if the system begins in some $|\alpha\rangle$ which is a superposition of the ground and excited states, it will remain in that state. So how can we calculate the transition probabilities in this case?


1 Answer 1


I'll assume the question refers to calculating coefficients $c_i(t)$, not transition probabilities under external interaction.

Since in Heisenberg representation all relevant quantities are calculated as observable/operator averages, the problem becomes one of choosing the correct observable to represent the final state. The simplest option is its projector, in this case either $P_e = |e\rangle\langle e|\;$ or $P_g = |g\rangle\langle g|$. For example, the average of the evolved projector $P^H_e(t) = \mathcal T e^{\frac{i}{\hbar}\int_0^t{d\tau H(\tau)}} P_{e}(0) \mathcal T e^{-\frac{i}{\hbar}\int_0^t{d\tau H(\tau)}}$ reads $$ \langle \alpha | P^H_{e}(t) | \alpha \rangle = \langle \alpha | \mathcal T e^{\frac{i}{\hbar}\int_0^t{d\tau H(\tau)}} P_{e}(0) \mathcal T e^{-\frac{i}{\hbar}\int_0^t{d\tau H(\tau)}} | \alpha \rangle = $$ $$ = \langle \alpha | \mathcal T e^{\frac{i}{\hbar}\int_0^t{d\tau H(\tau)}} |e\rangle\langle e| \mathcal T e^{-\frac{i}{\hbar}\int_0^t{d\tau H(\tau)}} | \alpha \rangle \equiv \langle \alpha_S(t) |e\rangle\langle e| \alpha_S(t) \rangle = |\langle e| \alpha_S(t) \rangle|^2 $$ where $| \alpha_S(t) \rangle = \mathcal T e^{-\frac{i}{\hbar}\int_0^t{d\tau H(\tau)}} | \alpha \rangle$ is the evolved state in the Schroedinger picture. The last expression obviously amounts to $|\langle e| \alpha_S(t) \rangle|^2 \equiv |c_e(t)|^2$, which is the desired transition probability. Similarly for $P^H_g(t)$ or any other state projector.


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