Question

Suppose \(\Psi(\mathrm{x}, \mathrm{y}, \mathrm{z})\) represents a particle in three dimensional space, then probability of finding the particle in the unit volume at a given point \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) is $\ldots \ldots$ (A) inversely proportional to $\Psi^{\prime}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ (B) directly proportional \(\Psi^{*}\) (C) directly proportional to \(\mid \Psi \Psi^{*}\) (D) inversely proportional to \(\left|\Psi \Psi^{*}\right|\)

Short answer

The probability of finding the particle in a unit volume at a given point \(x, y, z\) is directly proportional to \(|\Psi \Psi^{*}|\).

Step-by-step solution

Review basic concepts related to wave function

The wave function, \(\Psi(x, y, z)\), contains all the information about a particle's position in three-dimensional space. The probability of finding the particle in a particular position is related to the square of the magnitude of the wave function. So, we will need to find the correct expression for this probability using \(\Psi(x, y, z)\) and its given properties.

Calculate the probability density

The probability density, represented by \(P(x, y, z)\), is the probability of finding a particle per unit volume in a given point in space. To obtain the probability density, we multiply the wave function \(\Psi\) with its complex conjugate \(\Psi^{*}\). Thus, the probability density is given by \[P(x, y, z) = |\Psi(x, y, z) \Psi^{*}(x, y, z)|\]

Analyze the given options

Now that we have the probability density expression, we can analyze the given options: (A) inversely proportional to \(\Psi^{\prime}(x, y, z)\): This option implies that the probability density is indirectly related to the first derivative of the wave function. This is not correct as we have already found the correct expression above. (B) directly proportional to \(\Psi^{*}\): This option indicates that the probability density is proportional only to the complex conjugate of the wave function. However, we have seen that it is related to the product of the wave function and its complex conjugate, so this option is incorrect. (C) directly proportional to \(|\Psi \Psi^{*}|\): According to our expression for the probability density, this option is correct as it states that the probability density is directly proportional to the square of the magnitude of the wave function. (D) inversely proportional to \(|\Psi \Psi^{*}|\): This option implies that the probability density is inversely proportional to the square of the magnitude of the wave function, which is incorrect. So based on our analysis, the correct answer is (C) directly proportional to \(|\Psi \Psi^{*}|\).