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AccidentalFourierTransform
AccidentalFourierTransform
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Translator Translation Operator and Position Basis

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Qmechanic ♦
Qmechanic ♦
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In Modern Quantum MechanicsModern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $\left| x+dx\right\rangle \simeq \left| x \right\rangle$.$$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for every $\epsilon > 0$ $\langle x+ \epsilon \left| x \right\rangle = 0$.$$\langle x+ \epsilon \left| x \right\rangle = 0.$$ Therefore this limiting process $\lim_{\epsilon \rightarrow 0} \left| x+ \epsilon \right\rangle = \left| x \right\rangle$$$\lim_{\epsilon \rightarrow 0} \left| x+ \epsilon \right\rangle = \left| x \right\rangle$$ does not make sense for me. I couldn't derive commutator relation without using these. Thanks for any help.

In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $\left| x+dx\right\rangle \simeq \left| x \right\rangle$. But for every $\epsilon > 0$ $\langle x+ \epsilon \left| x \right\rangle = 0$. Therefore this limiting process $\lim_{\epsilon \rightarrow 0} \left| x+ \epsilon \right\rangle = \left| x \right\rangle$ does not make sense for me. I couldn't derive commutator relation without using these. Thanks for any help.

In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for every $\epsilon > 0$ $$\langle x+ \epsilon \left| x \right\rangle = 0.$$ Therefore this limiting process $$\lim_{\epsilon \rightarrow 0} \left| x+ \epsilon \right\rangle = \left| x \right\rangle$$ does not make sense for me. I couldn't derive commutator relation without using these.

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bronstein
bronstein
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Translator Operator

In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $\left| x+dx\right\rangle \simeq \left| x \right\rangle$. But for every $\epsilon > 0$ $\langle x+ \epsilon \left| x \right\rangle = 0$. Therefore this limiting process $\lim_{\epsilon \rightarrow 0} \left| x+ \epsilon \right\rangle = \left| x \right\rangle$ does not make sense for me. I couldn't derive commutator relation without using these. Thanks for any help.