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\begin{document}
\title{Title}
\author[1]{chaudhary148yagya}%
\affil[1]{Affiliation not available}%
\vspace{-1em}
\date{March 28, 2024}
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\par\null\par\null
\(H_H\left(t\right)\ =\ H_s\left(\hat p_H\left(t\right),\text{\hat x}_H\left(t\right),t\right)\)
\par\null\par\null
2. operator is time dependent (\(\bigcirc=\bigcirc\left(t\right)\))
\(H_H\left(t\right)=H_s\left(\hat p_H\left(t\right),\hat x_H\left(t\right),t\right)\)
which is contrasting to the Schr\selectlanguage{ngerman}ödinger picture where operators are
constant and state vectors are time dependent .~
In the Heisenberg picture , the basis does not change with time . This
is accomplished by adding a term to the~ Schrödinger states to eliminate
the time - dependence .
~~~~~~~~~~~~~~~~~~~~~~~ \(\left|\psi_H>=e^{\frac{iH_st}{\hbar}}\left|\psi_S\left(t\right)=\right|\psi_s\left(0\right)>\right|\)
The quantum operators , however do change with time .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\(O\ =O\left(t\right)\)
\textbf{Heisenberg Operators}
Let consider a Schrödinger operator~\(A_s\)(subscript s is
for Schrödinger). This operator may or may not have time dependence. Now
,we will examine~\(A_s\) in between time dependent states
\textbar{}a,t\textgreater{} and \textbar{}b,t\textgreater{} and~ use the
time-evolution operator to convert the states to time zero:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\(\ =\ \)
\(A_H\)(t) is defined as a Heisenberg operator associated
with~~\(A_s\) as the object in between the time equal zero
states :
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\(A_H\left(t,0\right)\ =\ U^+\left(t,0\right)A_sU\left(t,0\right)\)~~~~~~~~~~~
Now we will take some important consequences of the given definition :
~~~ 1. At t = 0 ~ Heisenberg operator becomes equal to Schrödinger
operator :
~ ~~~~~~~~~~~~~~~~~~~~ \(A_H\left(0\right)=A_s\)
~ 2. The Heisenberg operator associated with the product of Schrödinger
operators is equal to the~~~ ~~~~~~~~product of the corresponding
Heisenberg operators:
~~~~~~~~~~~~~~~~~~~~~~~~~~~\(C_S\ =\ A_SB_S\)~~~-\textgreater{}~~~~~\(C_H\left(t\right)=A_H\left(t\right)B_H\left(t\right)\)~~~~~~~~~
~~~~~~~~~~~~~~~~~~ \{ ~~~~~~~ \(C_H\left(t\right)=U^+\left(t,0\right)C_sU\left(t,0\right)\ =\ U^+\left(t,0\right)A_sB_sU\left(t,0\right)\)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ \(=U^+\left(t,0\right)A_sU\left(t,0\right)U^+\left(t,0\right)B_sU\left(t,0\right)\ =\ A_H\left(t\right)B_H\left(t\right)\)
~~~~~~~~~~ \}
3.~ ~~~~~~~
\par\null
\textbf{Heisenberg equation of motion~}
\textbf{c\(i\hbar\frac{\partial U\left(t,t_0\right)}{\partial t}\ =\ \ H_s\left(t\right)U\left(t,t_0\right)\)( 1)}
c\(i\hbar\frac{\partial U^+\left(t,t_0\right)}{\partial t}\) =~ \(-U^+\left(t,t_0\right)H_s\left(t\right)\) (2)
c\(i\hbar\frac{\partial A_H\left(t\right)\ }{\ \partial t}\) = ~\(\left(i\hbar\ \frac{\partial U^+\left(t,0\right)}{\ \partial t}\right)A_s\left(t\right)U\left(t,0\right)\) ~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~ + \(\)\(U^+\left(t,0\right)A_s\left(t\right)\left(i\hbar\frac{\partial U\left(t,0\right)}{\partial t}\right)\)
~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \(U^+\left(t,0\right)i\hbar\frac{\partial A_S\left(t\right)}{\partial t}U\left(t,0\right)\)
\(i\hbar\frac{\partial\ }{\partial t}A_H\left(t\right)\ =\ -U^+\left(t,0\right)H_s\left(t\right)A_s\left(t\right)U\left(t,0\right)\)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\(+\ U^+\left(t,0\right)A_s\left(t\right)H_s\left(t\right)U\left(t,0\right)\)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\(+\ \ U^+\left(t,0\right)i\hbar\frac{\partial A_s\left(t\right)}{\partial t}U\left(t,0\ \right)\)
\(i\hbar\frac{\partial A_{H\ }\left(t\right)}{\partial t}\ =\ -H_H\left(t\right)A_H\left(t\right)\ +\ A_H\left(t\right)H_H\left(t\right)\ +\ i\hbar\left(\frac{\partial A_s\left(t\right)}{\partial t}\right)_H\)
\(i\hbar\frac{\partial A_H\left(t\right)}{\partial t\ }\ =\ \left[A_H\left(t\right),H_H\left(t\right)\right]\ +\ i\hbar\left(\frac{\partial A_s\left(t\right)}{\partial t}\right)_H\)
c\(A_s:\ \left[A_s,\ H_s\right]\ =0\ \)~ \(\frac{\ \partial A_H\left(t\right)}{\partial t}=0\)
Comments~
1. \(i\hbar\frac{\partial A_H\left(t\right)}{\partial t}\ =\ \left[A_H\left(t\right),\ H_H\left(t\right)\right]\)
2.
\(i\hbar<\psi,t\left|A_s\right|\psi,t>\ =i\hbar<\psi,0\left|A_H\left(t\right)\right|\psi,0>\)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\(=\ <\psi,0\left|i\hbar\frac{\partial A_H\left(t\right)}{\partial t}\right|\psi,0>\)
= \(=<\psi,0\left|\left[A_H\left(t\right),H_H\left(t\right)\right]\right|\psi,0>\)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ imp
=\textgreater{} \(i\hbar\frac{\partial\ }{\partial t}\ =\ <\left[\left[A_H\left(t\right),H_H\left(t\right)\right]\right]>\)
\(i\hbar\frac{\partial\ }{\partial t}\ =\ <\left[\left[A_s,\ H_s\right]\right]>\) \textless{}- most imp\(\ \ \ \left[\ \ \vec{A}\ ,\ \vec{B}\ \ \right]\ =\vec{\ \ \ A.\vec{B}}\ -\ \vec{\ \ \ B.\ \vec{A}}\)
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