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Danijel Gorupec

Particle in a box; 'localized' integrals

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Hello... Say there is a particle in a 1D box. The particle is in certain state. I know its position-space wave function.

I can compute expectation value of its momentum by sandwiching the momentum operator between psi* and psi and integrating over the whole box width (or over whole infinity).

Question... If I don't integrate over the whole box, but only over a part of the box, can the result be interpreted as follows: if I simultaneously measure momentum and position of identically prepared setups, and if I only consider those cases where a particle was found within the part of the box for which I computed the integral, then the average momentum of those particles will equal to my integral?

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44 minutes ago, Danijel Gorupec said:

and if I only consider those cases where a particle was found within the part of the box for which I computed the integral,

This is your problem.

Please show an example of your intended calculations.

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1 hour ago, studiot said:

Please show an example of your intended calculations.

 

So, from various sources this is an time-independent wave-function for a ground state particle in an infinite square (widht 'a'). The momentum calculation gives zero, as expected.

particle_in_box.png.b04374a0f4a5203a9caa7d651b856dee.png

But if I only integrate over half of the box width (the last line) then I get an imaginary result. Does this mean that this result has no interpretation (that it, it means nothing)?

[sorry, the 'h' should actually be h-bar, but I don't have this symbol]

Edited by Danijel Gorupec

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Hmm... but let's compute expectation value for kinetic energy... I am doing this for the same wave-function (ground state particle-in-a-box):

('h' is actually h-bar)

particle_in_box_energy.png.5999e3ffbef619ed5237699237926a7a.png

The above is the correct result when the integral is computed over the whole box. But even if I integrate over only a part of the box, I still obtain real values - does it mean that such result might have some actual meaning?

Not being lazy, I tried to compute kinetic energy distribution (or density) over the box. Moderators can move this to the speculation forum.

I am using a sloppy notation as I know no better.

particle_in_box_energy2.png.cb1da923c9da1d1ea5212a1954039087.png

This is the kinetic energy density function for a ground-state particle in a 1D box, how I calculated it.... Can this have any meaning? The function shape resembles the position probability density function for the particle.

I would be very glad if there is any function that can show kinetic energy density for wave-like particles. I like energy distributed in space (that is, local i like the local conservation of it). However, I never heard anything about it.

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What is the point of doing the integral over only part of the spatial extent?

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