Which of the following is established under the postulates of quantum mechanics about the real wave function #Psi#, where #P# is the probability of finding a particle whose time-dependent state is described by #Psi# in volume #dV#?

The wave function #Ψ# is related to the probability of finding a particle in a given region of space.

The relationship is:

#P = int Ψ^2dV#

If we integrate #Ψ^2# over a given volume, we find the probability that the particle is in that volume.

If we integrate over all space, the probability must be 1.

#P = int_0^∞ Ψ^2dV = 1#

#P#, the probability density, is equal to #1#.

NORMALIZABILITY OF ANY WAVE FUNCTION

Two of the quantum mechanics postulates jointly say that the wave function must be orthonormal over allspace, so it must be normalized. Normalized means its probability density goes from #0%# to #100%#. Evidently, #100% = 1# if #0% = 0#.

If you are talking about spherical harmonics, then yes, allspace is basically #[0,oo)#.

#\mathbf(Psi(r,theta,phi) = R_(nl)(r)Y_(l)^(m)(theta,phi))#

where:

• #R_(nl)(r)# is the radial component of the wave function that describes distance from the center of a sphere outwards to infinity (#n# and #l# are the quantum numbers #n# and #l#)
• #Y_(l)^(m)(theta,phi)# is the angular component that may contain a dependence on #theta#, or may contain a dependence on both #theta# AND #phi#.

In that case, the radial component, #R_(nl)(r)#, assumes that domain of #[0,oo)#.

As a sidenote, a particle in a box model DOES show allspace as #(-oo,oo)#, where #-oo# is leftwards of the left-boundary #0# and #oo# is rightwards of the right-boundary #a#, where #a# is the length of the box whose height is #oo#.

CONDITIONS FOR A PHYSICALLY REALISTIC WAVE FUNCTION

For the time-dependent wave function #Psi# to be physically realistic, the following conditions are required:

• The wave function must be closed (it must have boundary conditions).
• It must be single-valued (i.e. it must pass the vertical line test).
• It must be continuous.
• #int_"allspace" Psi^"*"Psid tau = 1#, i.e. it must be normalizable over allspace.
• #int_"allspace" Psi_A ""^"*" Psi_Bd tau = 0#, i.e. two different wave functions must be orthogonal to each other.

ORTHONORMALITY OF THE 2P ORBITAL WAVE FUNCTIONS

For instance, #Psi_(2px)# and #Psi_(2py)#, the wave functions for the #2p_x# and #2p_y# atomic orbitals, are orthogonal and normalized. Therefore, they are known to be orthonormal (combining the two terms).

I've shown the normalizability of the #2p_z# atomic orbital here in case you want to see it (and be assured that there's no way you need to do it on a test):
http://socratic.org/scratchpad/02f319af3fee9184eb68

All that says is that the probability of finding an electron anywhere within a #2p_z# orbital is #100%#. Pretty intuitive. One can do a similar trick with the #2p_x# and #2p_y# and come to the same conclusion.

You can visually recognize the orthogonality by realizing that the x and y axes are perpendicular, and these atomic orbitals lie along those axes. Hence, their dot product is #hatx*haty = 0#, their cross product is #hatx xx haty = hatz#, and they are orthogonal.

I've also done a proof of the orthogonality here for the #2p_x# and #2p_y# atomic orbitals if you were curious:
http://socratic.org/scratchpad/d92580b37cb5a5249f5d