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We know the Schrodinger equation for free Hamiltonian is :

$$ i\hbar\frac{\partial\psi}{\partial t} = H_f \psi $$ the wave function could be written as $$ \psi(x,t)=\hat{S}(t) \psi(x,0) $$ $$ \hat{S}(t)=\exp\left(-\frac{i}{ \hbar }H_ft\right ) .....(1) $$ I want to know that how exponential term arisen in the equation (1) and how can we write the equation below . Please show me the matrix manipulation for the below equation .

$$ \hat{S}(t) = \tau_3 S^\dagger \tau_3 $$

Thanks .

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What exactly is $\tau_{3}$ –  Debangshu Oct 10 '12 at 11:14
    
τ3 Is pauli matrix here –  Unlimited Dreamer Oct 10 '12 at 12:52
    
See other Phys.SE question by OP for context. –  Qmechanic Oct 10 '12 at 22:51

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The first equation – in which you omitted $\psi$ on the left hand side – is a differential equation for $\psi$. The second equation reflects the fact that the equation is linear, so the evolution of $\psi$ after time $t$ will have to be linear as well, and it is given by an operator called $S(t)$ that stands on the left side from $\psi$, the most general linear operator acting on state vectors.

The particular form of $S(t)$ in the third equation is a solution to the differential equation, i.e. the first equation you wrote. Solutions to differential equations can't always be "derived in a straightforward way" and even if they can be solved, there may always be some kind of a guess. Anyway, everyone familiar with basics of calculus sees why this $S(t)$ solves the differential equation at the top. The $t$-derivative of the exponential is once again the exponential times the $t$-derivative of the argument of the exponential, and the derivative of the argument with respect to $t$ is clearly $-i H_f/ \hbar$ which is the right factor that jumps out in the first equation.

Concerning the $\tau_3$ equation, you would have to explain what you exactly mean by $\tau_3$ and what's the argument of $S^\dagger$. In general, however, the equation says that $S$ and $S^\dagger$ are equivalent. What "matrix manipulation" do you exactly want to see?

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I understand the solution S(t) $$ \tau_3 $$ Is pauli matrix here –  Unlimited Dreamer Oct 10 '12 at 12:00
    
If it were just a 2-dimensional Pauli matrix, it would mean that $S$ has to act on the same 2-dimensional space, so it would be a rather simple $S$. You either have to give us the context or your question is meaningless as stated. There are surely some interpretations of the matrix for which the formula is true and it may also be true in various 2-level systems but I am not gonna guess what you meant. –  Luboš Motl Oct 11 '12 at 17:33

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