Where Is a Particle Most Likely Found in Quantum Mechanics?

[NeedPhysHelp8]

Homework Statement

a particle is described by the normalized wave function
$$\psi(x,y,z) = Axe^{-\alpha x^2}e^{-\beta y^2}e^{-\gamma z^2}$$
Where all constants are positive and real. The probability that the particle will be found in the infinitesimal volume dxdydz centered at point $$(x_{0},y_{0},z_{0})$$ is $$\mid \psi (x_{0},y_{0},z_{0}) \mid ^2 dxdydz$$

a) at what values of $$x_{0}$$ is the particle most likely to be found
b) are there any values of x for which the probability of the particle being found is zero?explain

Homework Equations

Alright so I am a newb when it comes to QM because we're just learning it now, I'm very confused with this question because it asks for probability of x when its over a region of x,y and z. Is it possible to use this $$\int \mid \psi (x_{0},y_{0},z_{0}) \mid ^2 dxdydz$$ and integrate over all space?? please can someone tell me where to start

 

[chrispb]

The question is kind of poorly worded. If you integrate |psi|^2 over all space, you should get 1 (the probability of finding the particle anywhere is 1), and that let's you fix the constant A in terms of alpha, beta and gamma. What they probably mean by (a) is: for a fixed value of y and z, the neighborhood around what value of x maximizes your probability of finding the particle? In (b), they mean the same sort of thing: are there any regions of space where the particle won't be?

 

[NeedPhysHelp8]

ok, but I still don't understand what to do...please someone just tell me where to start i'd really appreciate it this is frustrating me so much

 

[chrispb]

For what value(s) of x is |psi|^2 at a maximum? For what value(s) of x is |psi|^2 0?

 

[vela]

As chrispb noted, the question is a bit ambiguous. What I think they want you do to is find the marginal probability density p_x(x) and find where it's a maximum and where it's zero. To find p_x(x), you integrate over y and z, so you're just left with x as a variable.

chrispb suggested the other way to interpret the question. It turns out you'll get the same answer either way because of the wave function you have.

 

[NeedPhysHelp8]

oh ok, so i integrate over all space for y and z then I'm left with x and just solve for it then? i'll give er a try