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Qmechanic ♦
Qmechanic ♦
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So I’m doing QM and found bunch of problems for beginners and I’m struggling with this one: $\lim_{a\rightarrow 0}\int^{\infty}_{-\infty}e^{\frac{ip x}{\hbar}-a x^2}dx=2\pi\hbar\delta(p).$$$\lim_{a\rightarrow 0}\int^{\infty}_{-\infty}e^{\frac{ip x}{\hbar}-a x^2}dx=2\pi\hbar\delta(p).$$ If I swap the limit and the integral I get the correct answer, but can I do that?

So I’m doing QM and found bunch of problems for beginners and I’m struggling with this one: $\lim_{a\rightarrow 0}\int^{\infty}_{-\infty}e^{\frac{ip x}{\hbar}-a x^2}dx=2\pi\hbar\delta(p).$ If I swap the limit and the integral I get the correct answer, but can I do that?

So I’m doing QM and found bunch of problems for beginners and I’m struggling with this one: $$\lim_{a\rightarrow 0}\int^{\infty}_{-\infty}e^{\frac{ip x}{\hbar}-a x^2}dx=2\pi\hbar\delta(p).$$ If I swap the limit and the integral I get the correct answer, but can I do that?

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MZperX
MZperX
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Quantum mechanics Dirac delta representation with integral

So I’m doing QM and found bunch of problems for beginners and I’m struggling with this one: $\lim_{a\rightarrow 0}\int^{\infty}_{-\infty}e^{\frac{ip x}{\hbar}-a x^2}dx=2\pi\hbar\delta(p).$ If I swap the limit and the integral I get the correct answer, but can I do that?