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- Thread starter vincent_vega
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TSny

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Hello, vincent.

You have a good question there. It is possible to construct functions ψ(x) that go to zero at infinity but such that ψ^{*}(x)ψ'(x) is not defined at infinity. For example ψ = sin[itex](x^4)/\sqrt{1+x^2}[/itex]

Griffiths wants to argue that the combination ψ^{*}(x)ψ'(x) - ψ(x)ψ^{*}'(x) goes to zero at infinity so that the normalization of the wavefunction is time independent. For any *real* wavefunction, you can see that this combination is identically zero for all x. However, it is possible to construct complex valued functions ψ for which the combination does not go to zero at infinity even though ψ goes to zero at infinity. For example ψ = [itex]e^{ix^4}/\sqrt{1+x^2}[/itex]. But this wavefunction is "pathological". For example, the expectation value of the kinetic energy operator is undefined for this function.

So, maybe the allowable wavefunctions are restricted to exclude these pathological functions. I don't know.

You have a good question there. It is possible to construct functions ψ(x) that go to zero at infinity but such that ψ

Griffiths wants to argue that the combination ψ

So, maybe the allowable wavefunctions are restricted to exclude these pathological functions. I don't know.

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Hello, vincent.

You have a good question there. It is possible to construct functions ψ(x) that go to zero at infinity but such that ψ^{*}(x)ψ'(x) is not defined at infinity. For example ψ = sin[itex](x^4)/\sqrt{1+x^2}[/itex]

Griffiths wants to argue that the combination ψ^{*}(x)ψ'(x) - ψ(x)ψ^{*}'(x) goes to zero at infinity so that the normalization of the wavefunction is time independent. For anyrealwavefunction, you can see that this combination is identically zero for all x. However, it is possible to construct complex valued functions ψ for which the combination does not go to zero at infinity even though ψ goes to zero at infinity. For example ψ = [itex]e^{ix^4}/\sqrt{1+x^2}[/itex]. But this wavefunction is "pathological". For example, the expectation value of the kinetic energy operator is undefined for this function.

So, maybe the allowable wavefunctions are restricted to exclude these pathological functions. I don't know.

thanks that makes sense.

do you have any idea why x*ψ(x)*ψ'(x) is defined to be zero at x = infinity? this also occurred in the example by griffiths. I am thrown off by the x in front of the expression

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TSny

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thanks that makes sense.

do you have any idea why x*ψ(x)*ψ'(x) is defined to be zero at x = infinity? this also occurred in the example by griffiths. I am thrown off by the x in front of the expression

I don't know. It seems that Griffiths is making an assumption about the behavior of the wavefunction and its derivative at infinity.

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