91Ó°ÊÓ

In quantum mechanics, the concept of wave function normalization is crucial. This ensures that the total probability of finding a particle somewhere in space is equal to one. The wave function, denoted as \( \psi(x) \), needs to satisfy the integral condition \( \int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 1 \). This implies that all probabilities calculated using \( \psi(x) \) are valid and meaningful.

For the given wave function \( \psi(x) = b^{-\frac{1}{2}}|x / b|^{\frac{1}{2}} e^{-(x / b)^{2} / 2} \), the normalization process involves squaring the modulus (the absolute value) of the wave function. This transforms the expression to \( b^{-1} (x^2/b^2)^{\frac{1}{2}} e^{-(x/b)^2} \).

By changing variables such as \( y = \frac{x}{b} \), the integral simplifies to \( \int_{-\infty}^{\infty} |y| e^{-y^2} \, dy = \sqrt{\pi} \). This integral result ensures the function is normalized, as multiplying \( \sqrt{\pi}b^{-1} \) aligns it with the condition \( \int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 1 \). Hence, for the given wave function at \( b = 1.0 \text{ nm} \), we find it correctly normalized.

In quantum mechanics, finding the most probable position of a particle described by a wave function involves determining where \(|\psi(x)|^2\) — interpreted as a probability density — is maximized. For regions in space where there is the highest likelihood of locating a particle, you generally compute the derivative of \( |\psi(x)|^2 \) and set it to zero to locate the critical points.

For the wave function \( \psi(x) = b^{-\frac{1}{2}}|x / b|^{\frac{1}{2}} e^{-(x / b)^{2} / 2} \) in the positive region \( x>0 \), the task boils down to maximizing \( b^{-1} (x^2/b^2)^{\frac{1}{2}} e^{-(x/b)^2} \). Taking its derivative \( \frac{d}{dx}\left(x e^{-x^2/(2b^2)}\right) \) and setting the resulting equation \( b - \frac{x^2}{b} = 0 \) leads directly to the most probable position.

Solving yields that \( x = b \), which given \( b = 1.0 \text{ nm} \), places the most probable position at \( x = 1.0 \text{ nm} \). This result is profoundly aligned with solving and verifying the condition for peak probability density.

Calculating the probability of finding a particle within a specified region using a quantum mechanical wave function involves integrating the probability density \( |\psi(x)|^2 \) over that region. For instance, the probability of locating the particle between \( x=0 \) and \( x=0.5 \text{ nm} \) requires computing \( \int_{0}^{0.5} |\psi(x)|^2 \, dx \).

For the given wave function \( \psi(x) = b^{-\frac{1}{2}}|x / b|^{\frac{1}{2}} e^{-(x / b)^{2} / 2} \), the integral simplifies to \( \int_{0}^{0.5} b^{-1} (x/b) e^{-(x/b)^2} \, dx \).

Using a substitution \( y = \frac{x}{b} \), which makes integrating easier, yields \( P = \frac{1}{2}\left(1 - e^{-1/4}\right) \). This calculation results in an approximate probability of 0.1175, indicating a non-negligible likelihood of finding the particle within the defined boundaries of \( x=0 \) to \( x=0.5 \text{ nm} \). This step underscores one essential quantum aspect where probabilities are inherently expressed in ranges instead of pinpoint precision.