Wave Functions increase in magnitude over time ?

[Widdekind]

Please consider the Hydrogen 1S ground-state Wave Function (WF), [math]\propto \frac{e^{-r}}{r}[/math]. If that WF is normalized, so that its modulus squared integrates to one, [math]\int r^2 \times \frac{e^{-2r}}{r^2} dr \rightarrow 1[/math], then the WF itself integrates to two, [math]\int r^2 \times \frac{e^{-r}}{r} dr = 2[/math].

 

(Of course, that 'direct WF integral' is only the amplitude, to be multiplied by the appropriate complex exponential of the ground-state energy-frequency, [math]\propto e^{-13.7 \, eV \, \times t}[/math].)

 

QUESTION: For a general 'wave packet' -- which 'spreads through space', over time, according to its evolution, as described by the SWE -- would not the 'direct integral', of the WF itself, actually increase over time ?? For, whilst a highly spatially concentrated WF, and its square, aren't dramatically different -- [math]x \approx x^2[/math] when [math]x \approx 1 \, or \, 0[/math] -- conversely, a low-and-flat, spread-out WF, will be much higher in absolute value, than its square. If so, then the spreading of the WF, actually increases the "overall integrated aggregate amount", of WF present, even if its squared-modulus remains normalized.

 

If so, then, in some sense the SWE "grows" the WF, over time, even though the WF² remains normalized to one.

[Widdekind]

Please consider the Hydrogen 1S ground-state Wave Function (WF), [math]\propto \frac{e^{-r}}{r}[/math]. If that WF is normalized, so that its modulus squared integrates to one, [math]\int r^2 \times \frac{e^{-2r}}{r^2} dr \rightarrow 1[/math], then the WF itself integrates to two, [math]\int r^2 \times \frac{e^{-r}}{r} dr = 2[/math].

 

Oops -- [math]\int_0^{\infty} r^2 \times \frac{e^{-2r}}{r^2} dr = \frac{1}{2} \rightarrow 1[/math], so that our probability density has an extra factor of two (2), and so our WF has an extra factor of square-root-two (2^1/2). And, so, the spatial integral, of the original WF itself, integrates to [math]\int_0^{\infty} r^2 \times \sqrt{2} \frac{e^{-r}}{r} dr = \sqrt{2}[/math]. Thus, the WF spatially integrates to square-root-two (2^1/2) times that of the probability density.

 

 

 

 

Now, this same "square-root-two (2^1/2) result" appears to be the case, for a spreading Gaussian wave-packet, as well. For, according to Pan 2009, a spreading Gaussian WF has the form:

 

[math]\Psi \propto \frac{1}{\sqrt{1 + i t}} e^{-\frac{x^2}{1+i t}} = \frac{1}{\sqrt{1 + i t}} e^{-\frac{x^2}{1+ t^2} \times (1 - i t)} [/math]

Substituting [math]u \equiv \frac{x}{\sqrt{1+t^2}}[/math], that integral computes to:

 

[math]\frac{\sqrt{1+t^2}}{\sqrt{1+it}} \int_{-\infty}^{\infty} du \; e^{-u^2 \times (1 - i t)} = \frac{\sqrt{1+t^2}}{\sqrt{1+it}} \times \frac{\sqrt{\pi}}{\sqrt{1 - i t}} = \sqrt{\pi}[/math]

according to an online integrator. And, thus, the probability density, [math]\Psi^{*} \Psi[/math], integrates to:

 

[math]\frac{\sqrt{1+t^2}}{\sqrt{1+it} \times \sqrt{1 -it}} \int_{-\infty}^{\infty} du \; e^{-u^2 \times 2} = \frac{\sqrt{\pi}}{\sqrt{2}}[/math]

Again, the WF spatially integrates to "square-root-two (2^1/2) times" that of the associated, squared-modulus, probability density (PD).

 

CONCLUSION ??

 

The WF has a "constant mathematical reality", which underlies that of the associated, squared-modulus, PD. The WF is "mathematically conserved", in overall, aggregate, spatially-integrated value. And, not implausibly, the WF may generally integrate to "square-root-two (2^1/2) times" as much, as its associated PD.

[swansont]

Wave functions can also integrate to zero, or negative values. They are not associated with anything physical. "constant mathematical reality" is kind of meaningless. The spatial wave function is time independent, by definition. That's a tautology.

[Widdekind]

According to the cited source, WFs, initially confined to a spatial size-scale [math]l_0[/math], at an (asymptotic) speed, normalized to that of light, of [math]\beta \approx \frac{1}{2} \left( \frac{l_0}{\lambda_C} \right)^{-1}[/math]. This, of course, comports with the HUP, according to which [math]\Delta p \approx \frac{\hbar}{2} \Delta x^{-1}[/math], dividing through by the mass m, and then speed-of-light c, and converting the quantity [math]\frac{\hbar}{c m} \rightarrow \lambda_C[/math]. Thus, when quantum particles are spatially compressed, to their "Compton" size-scales, Relativistic effects begin to become important.