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1

Comment to the post (v2): It is true that if $t\mapsto q(t)$ is a solution to the Euler-Lagrange (EL) eqs. for the action $$S[q]~=~\int_{t_i}^{t_f} \! dt L(q^1,q^2,q^3, \ldots; (\dot{q}^1)^2,(\dot{q}^2)^2,(\dot{q}^3)^2, \ldots)$$ of a time-symmetric Lagrangian, then the time-reversed path $t\mapsto q(-t)$ would also be a solution to the EL eqs. In fact, ...

2

A time reversal operator is an anti-unitary operator, which can be expressed as: $\mathcal{T}=UK$ where $K$ denotes complex conjugate and $U$ is a unitary operator. In case of spinless particles, $U$ is chosen to be Identity. Thus $\mathcal{T}=K$. If the system has $\mathcal{T}$-reversal symmetry: $$KH\psi=HK\psi$$ which leads to:  ...

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I believe Shankar actually defines the quantum time reversal operator to be complex conjugation in the coordinate basis like as follows: 〈x|T|ψ〉=ψ*(x) Check 〈x|TXT|ψ〉=〈x|X|ψ〉 and 〈x|TPT|ψ〉=〈x|-P|ψ〉 Note that T is self-adjoint, so TXT and TPT express transformations of X and P into their time reversed forms.

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To understand my answer, you have to imagine space as a 2-D plane and time as a vector perpendicular to it. "Normal" time is defined as a vector pointing to the "right." If time is now instantaneously reversed (vector now pointing to the left), nothing would change (laws of physics, etc.), because we could just as easily define "normal" time as pointing to ...

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Is time reversible? Look at the stroboscopic photograph. Is the ball "falling up" or falling down? The answer is surely we don't know! A motion picture of this sort of sequence of the event could be run backward & would inevitably be impossible for the viewer to detect any violation of Newton's laws. A time-reversal changes both $t,v \to -t,-v$ ...

1

To expand a bit on Ian's answer, let's consider the case of a ball that has dropped from a given height and has bounced repeatedly against the ground, losing energy through partially inelastic collision with the ground until all its energy is dissipated as heat. Now if at this point time was reversed, what we might expect to see is that the flow of entropy ...

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Good question. Let's first consider the ball falling immediately before it hits the table. Neglect friction with the air for simplicity. The ball has a velocity in the downward direction. If we reverse time, the ball is in the same position right above the table, but now it has an upward velocity. A ball with an upward velocity will rise with a negative ...

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At present, the mechanism of the superconductivity in systems like Na$_{0.35}$CoO$_2$·1.3H$_2$O are of high interest. The CoO$_2$ layers have been modeled as a spin 1/2 antiferromagnetic Mott insulator on a triangular lattice. By using resonate valence bond mean-field analysis, this supports the view that the superconducting order parameter has the ...

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