Normalize the wave function:
Ψ (x) = C / (a^2 + x^2)
and find the expectation values of x and x^2 for this wave function.
Can someone point me in the right direction?
Ψ (x) = C / (a^2 + x^2)
and find the expectation values of x and x^2 for this wave function.
Can someone point me in the right direction?
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Assuming we talk about quantum mechanics, the idea is that the quantitiy |Psi(x)|^2 is the probability density function for finding the particle at location x.
Since the probability to find the particle somewhere at all must be exactly one, this means
Integral over |Psi(x)|^2 for x from -infinity to infinity must be one. So what you have to do is calculate the integral from -infinity to infinity over C^2/ (a^2 + x^2)^2.
I'd look up that integral in a table somewhere as it is a bit tedious to carry it out by hand.
Let's say the value of that integral is X. Then the normalization factor for Psi(x) will be 1/sqrt(X).
To compute the expectation values x and x^2, you do the integral from -infinity to infinity over x * |Psi(x)|^2 or x^2 * |Psi(x)|^2, respectively.
Since the probability to find the particle somewhere at all must be exactly one, this means
Integral over |Psi(x)|^2 for x from -infinity to infinity must be one. So what you have to do is calculate the integral from -infinity to infinity over C^2/ (a^2 + x^2)^2.
I'd look up that integral in a table somewhere as it is a bit tedious to carry it out by hand.
Let's say the value of that integral is X. Then the normalization factor for Psi(x) will be 1/sqrt(X).
To compute the expectation values x and x^2, you do the integral from -infinity to infinity over x * |Psi(x)|^2 or x^2 * |Psi(x)|^2, respectively.