Question

The Hamiltonian of a quantum system has the form $H = H_0 + H_1$. The so-called interaction (or Dirac) picture is obtained by making the following transformation $\left| \psi_I(t) \right> = \exp\left[ \frac{i}{\hbar} tH_0 \right] \left| \psi(t) \right>$, $\hat{A}_I(t) = \exp\left[ \frac{i}{\hbar} tH_0 \right] \hat{A} \exp\left[ -\frac{i}{\hbar} tH_0 \right]$, where $\left| \psi(t) \right>$ and $\hat{A}$ represent the state vector and an operator in the Schrodinger picture. a) Find the equations of motion for $\left| \psi_I(t) \right>$ and $\hat{A}_I(t)$. b) Show that the time dependence of the expectation value $<A>$ of the observable represented by the operator $\hat{A}$ is the same in the Dirac and Schrodinger pictures.

          The Hamiltonian of a quantum system has the form $H = H_0 + H_1$. The so-called interaction (or Dirac) picture is obtained by making the following transformation
$\left| \psi_I(t) \right> = \exp\left[ \frac{i}{\hbar} tH_0 \right] \left| \psi(t) \right>$,
$\hat{A}_I(t) = \exp\left[ \frac{i}{\hbar} tH_0 \right] \hat{A} \exp\left[ -\frac{i}{\hbar} tH_0 \right]$,
where $\left| \psi(t) \right>$ and $\hat{A}$ represent the state vector and an operator in the Schrodinger picture.
a) Find the equations of motion for $\left| \psi_I(t) \right>$ and $\hat{A}_I(t)$.
b) Show that the time dependence of the expectation value $<A>$ of the observable represented by the operator $\hat{A}$ is the same in the Dirac and Schrodinger pictures.

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The Hamiltonian of a quantum system has the form H = H0 + H1. The so-called interaction (or Dirac) picture is obtained by making the following transformation
| (t) > = [ (i)/(ħ) tH0 ] | ψ(t) >,
ÂI(t) = [ (i)/(ħ) tH0 ] Â[ -(i)/(ħ) tH0 ],
where | ψ(t) > and Â represent the state vector and an operator in the Schrodinger picture.
a) Find the equations of motion for | (t) > and ÂI(t).
b) Show that the time dependence of the expectation value <A> of the observable represented by the operator Â is the same in the Dirac and Schrodinger pictures.

Added by Ana T.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Adi S

06/21/2023

Step 1

Step 1: To find the equations of motion for |ψ_I(t)> and A_I(t), we start by applying the time derivative to the transformation equation: d/dt |ψ_I(t)> = d/dt (e^(tH_0) |ψ(t)>) Show more…

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The Hamiltonian of a quantum system has the form H=H_(0)+H_(1). The so-called interaction (or Dirac) picture is obtained by making the following transformation |psi _(I)(t): where |psi (t): and hat(A) represent the state vector and an operator in the Schrodinger picture. a) Find the equations of motion for |psi _(I)(t): and hat(A)_(I)(t). b) Show that the time dependence of the expectation value (:A:) of the observable represented by the operator hat(A) is the same in the Dirac and Schrodinger pictures. The Hamiltonian of a quantum system has the form H = Ho + H. The so-called interaction or Dirac) picture is obtained by making the following transformation |31(t)) =exp tHo|(t)) h where |(t)) and A represent the state vector and an operator in the Schrodinger picture. a) Find the equations of motion for (r(t)) and Ar(t). b) Show that the time dependence of the expectation value (A of the observable represented by the operator A is the same in the Dirac and Schrodinger pictures.

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