Heisenberg Picture In the Schro — dinger approach to quantum mechanics, operators O ‘ are time-independent and the wave function of the system carries all the time-dependent information. The expectation value of O ‘ obeys the Ehrenfest relation i d XO ‘ \ dt = ZBO ‘ , HF^ Buried in this is the time-dependent wavefunction ΨHt L\ =ª - Ht ΨH0L] = UHt L ΨH0L] If we make this substitution into an expectation value we get XΨHt L O ΨHt L\ = YΨH0L U -1 Ht L OUHt L ΨH0L] ” XΨH0L OHt L ΨH0L\ where in this Heisenberg picture the (time-independent) Schro — dinger operator O becomes time-depen dent: OHt L = U -1 Ht L OUHt L The Heisenberg operator OHt L obeys an equation similar to the Ehrenfest equation, but without the expectation values: (1) dO ‘ Ht L dt = BO ‘ Ht L, HF Such an equation is understood to operate on a time-independent wavefunction Ψ 0 \ = ΨH0L\. An example of the usefulness of this is to discover the operator corresponding to the vector potential A x of a single mode of the electromagnetic field, given the electric field operator E x Ht L = EIaHt L + a † Ht LM sinHkzL =- dA x Ht L dt Comparing this to Eq. (1) gives (2) dA x Ht L dt = - @A x Ht L, HD =-EIaHt L + a † Ht LM sinHkzL Since @a, HD = Ω a, Aa † , HE =- Ω a † , we get daHt L dt = - Ω aHt L so aHt L = a ª - Ω t , a † Ht L = a † ª Ω t so E x Ht L = EIa ª - Ω t + a † ª Ω t M sinHkzL =- dA x Ht L dt so A x Ht L = - E Ω Ia ª - Ω t - a † ª Ω t M sinHkzL so the correct Schrodinger vector potential operator is A x = A x H0L = - E Ω Ia - a † M sinHkzL