Chapter 41, Problem 41.1DQ

To determine

Compare the probability of finding the particle $A$ described by the wave function $\psi \left(x,y,z\right)$ and $B$ described by the wave function $\psi \left(x,y,z\right)e^{i\varphi }$ within a volume $dV$.

Answer to Problem 41.1DQ

The probability of finding the particle $A$ described by the wave function $\psi \left(x,y,z\right)$ is same as that of $B$ described by the wave function $\psi \left(x,y,z\right)e^{i\varphi }$ within a volume $dV$.

Explanation of Solution

Probability of finding the particle is known as probability density of the particle.

Write the expression for the probability density of the particle in volume $dV$

${\left|\psi \left(x,y,z\right)\right|}^{2}dV={\psi }^{*}\left(x,y,z\right)\psi \left(x,y,z\right)dV$ (1)

Here,

$\psi \left(x\right)$ is the wave function of particle.

$dx$ is a small distance in $x$ axis

Substitute $\psi \left(x,y,z\right)$ in equation (1) to get probability of finding the particle $A$ in volume $dV$.

$\begin{array}{c}{P}_A={\left|\psi \left(x,y,z\right)\right|}^{2}dV\\ ={\psi }^{*}\left(x,y,z\right)\psi \left(x,y,z\right)dV\end{array}$

Here,

$P_A$ is the probability density of particle $A$ in volume $dV$.

Substitute $\psi \left(x,y,z\right)e^{i\varphi }$ in equation (1) to get probability of finding the particle $A$ in volume $dV$.

$\begin{array}{c}{P}_B={\left|\psi \left(x,y,z\right)e^{i\varphi }\right|}^{2}dV\\ ={\left(\psi \left(x,y,z\right)e^{i\varphi }\right)}^{*}\left(\psi \left(x,y,z\right)e^{i\varphi }\right)dV\\ ={\psi }^{*}\left(x,y,z\right)\psi \left(x,y,z\right)e^{-i\varphi }{e}^{i\varphi }dV\\ ={\psi }^{*}\left(x,y,z\right)\psi \left(x,y,z\right)dV\end{array}$

Here,

$P_B$ is the probability density of particle $B$ in volume $dV$.

Thus from above two calculation the probability of finding particle $A$ in volume $dV$ is same as that of $B$ .

Conclusion:

Thus, the probability of finding the particle $A$ described by the wave function $\psi \left(x,y,z\right)$ is same as that of $B$ described by the wave function $\psi \left(x,y,z\right)e^{i\varphi }$ within a volume $dV$.