I'm trying to calculate this limit to answer a question in Quantum Mechanics:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\mathop {\lim }\limits_{{t_1} \to 0} \,\,{\left( {\frac{m}{{2\pi \hbar i{t_1}}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle 2$}}}}{e^{im{{(x' - x)}^2}/2\hbar {t_1}}}\,\,\,\,\, + \,\,\,\,\,\mathop {\lim }\limits_{{t_2} \to 0} \,\,{\left( {\frac{m}{{2\pi \hbar ( - i){t_2}}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle 2$}}}}{e^{ - im{{(x - x')}^2}/2\hbar {t_2}}}[/tex]

It seems ast→ 0 in an arbitrary way, the complex exponentials circles wildly from +1 toito -1 to -ito +1 again. And the two square roots approach ∞ in magnitude and are 90° out of phase with each other. So it seems the limit does not approach any particular value, not even to plus or minus ∞; the limit seem undefined.

However, I wonder ift_{1}andt_{2}could approach 0 in some controlled way that allows the two terms to cancel out.

Let

[tex]A = {\left( {\frac{m}{{2\pi \hbar i}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle 2$}}}}[/tex]

And let,

[tex]B = m{(x' - x)^2}/2\hbar [/tex]

Then the above limit can be written,

[tex]A\left( {\mathop {\lim }\limits_{{t_1} \to 0} \frac{{{e^{iB/{t_1}}}}}{{{t_1}^{{\raise0.5ex\hbox{$\scriptstyle 1$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle 2$}}}}} + i\mathop {\lim }\limits_{{t_2} \to 0} \frac{{{e^{ - iB/{t_2}}}}}{{{t_2}^{{\raise0.5ex\hbox{$\scriptstyle 1$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle 2$}}}}}} \right)[/tex]

which equals,

[tex]A\left( {\mathop {\lim }\limits_{{t_1} \to 0} \frac{{{e^{iB/{t_1}}}}}{{{t_1}^{{\raise0.5ex\hbox{$\scriptstyle 1$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle 2$}}}}} + \mathop {\lim }\limits_{{t_2} \to 0} \frac{{{e^{ - iB/{t_2} + i\pi /2}}}}{{{t_2}^{{\raise0.5ex\hbox{$\scriptstyle 1$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle 2$}}}}}} \right)[/tex]

If we restrictt_{2}such that,

[tex]\frac{{ - iB}}{{{t_2}}} + \frac{{i\pi }}{2} = \frac{{iB}}{{{t_1}}} + i\pi - i2\pi n[/tex]

fornany integer, then the [itex]i\pi - i2\pi n[/itex] factor in the exponent will insure that the second term is always 180° out of phase with the first term so that the two terms will cancel out to zero. In this case,t_{1}could be any arbitrary number approaching zero. Butt_{2}would have to be

[tex]{t_2} = \frac{B}{{\frac{{ - B}}{{{t_1}}} + 2\pi (n - \frac{1}{4})}}[/tex]

We can see from this thatt_{2}gets arbitrarily close to zero from above or below asnincreases to plus infinity or negative infinity, respectively. It seems that for any other way of lettingt_{2}approach zero, the limit is completely undefined.

The question is whether it is allowed to let parameters be discrete in the limiting process. Or must they always be continuous?

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# A Taking limits in discrete form

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