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Is there any relationship between the probability density in QM given by \begin{equation} P\left(t\right) = \int\limits_{V} \psi^{*}(\mathbf{x},t)\psi(\mathbf{x},t)\mathrm{d}^{3}\mathbf{x} \end{equation} and the mass $m$ of the particle? $\psi\left(\mathbf{x},t\right)$ is the wave function of the particle.

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    $\begingroup$ Be very careful when using $r$ as variable: if you integrate over the radius (and not $x,y,z$) then you must have the Jacobian of the transformation in the integral $r^2 dr$. $\endgroup$ – gented Jul 25 '17 at 22:58
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In general, no. However if you assume particular dynamics/Hamiltonian (e.g. a free particle), then you can just solve for the eigenfunctions directly and get a formula that involves your boundary condition and the mass.

Also, if you integrate this quantity over all of space, it must be one by normalization.

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Your integral is not the probability density. Depending on the limits of the volume, it is the probability of the particle being in that volume. If you integrate over all space, $P(t)=1$. If you integrate over some other volume, $P(t)\le 1$.

The probability density is $\psi ^*(\vec{r},t),\psi (\vec{r},t)d^3 r$, the integrand of that integral. Because $\psi(\vec{r},t)$ for a particular potential may depend on the particle mass, the probability density may, too, although probably in some non-linear way.

Consider the one-dimensional infinite square well of width $a$ bounded at $x=0$ and $x=a$. The time-dependent eigenfunctions are $$\psi_n(x,t)=\sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right)e^{(-iE_n t/\hbar)}$$ where $$E_n=\frac{n^2\pi^2\hbar^2}{2ma^2},$$ and $n$ is some positive integer.

If the particle is in a pure eigenstate, then the probability density will be independent of the mass. If the particle is in a mixed state, $$\psi(x,t)=\sum_{n=1}^{\infty} C_n\psi_n(x,t)$$ $$ \sum |C_n|^2=1$$ the probability density will have the mass in the product of the exponentials. If the boundaries of the integral are not $0\to a$, then the mass will appear in the probability calculation, otherwise it vanishes.

For the quantum harmonic oscillator, the mass of the particle appears in several terms of $\psi$, including the normalization constant (to the 1/4 power), so the mass will affect the probability density in a complicated, nonlinear fashion.

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