91Ó°ÊÓ

Normalization is a crucial step in quantum mechanics to ensure that the probability of finding a particle somewhere in space is equal to one. For the given wave function \( \Psi(x, 0) = A e^{-a|x|} \), we need to determine the appropriate value of \( A \).
The normalization condition requires us to set the integral of the square modulus of the wave function over all space equal to one:

• \( \int_{-\infty}^{\infty} |\Psi(x, 0)|^2\, dx = 1 \)

This involves solving:

• \( A^2 \int_{-\infty}^{\infty} e^{-2a|x|} \ dx = 1 \)

After splitting and evaluating the integral for each half of the domain due to the absolute value, we find:

• \( A^2 \cdot \frac{1}{a} = 1 \)
• \( A = \sqrt{a} \)

Through this process, we derive that the normalization constant \( A \) must be \( \sqrt{a} \), ensuring our wave function is properly normalized.

The Fourier transform is a powerful mathematical tool used to transform a wave function from position space to momentum space. This process is essential for analyzing how wave properties like momentum distribute across different states. For our wave function \( \Psi(x, 0) = \sqrt{a} e^{-a|x|} \), performing the Fourier transform gives us \( \phi(k) \), the wave function in momentum space.
The Fourier transform can be expressed as:

• \( \phi(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \Psi(x, 0) e^{-ikx} \ dx \)

After splitting and solving the integral, we find:

• \( \phi(k) = \frac{\sqrt{a}}{\sqrt{2\pi}} \frac{2a}{a^2 + k^2} \)

This result indicates how different momentum components are weighted, illustrating the distribution of momentum for the particle.

Momentum space offers a different perspective from position space, focusing on the momentum components of a wave function rather than their spatial distribution. The concept helps us to understand phenomena like diffraction and quantum states more deeply.
For the Fourier transform result \( \phi(k) \) derived earlier, momentum space provides insight into how a particle's probability is distributed across various momentum values. The expression for \( \phi(k) \):

• \( \phi(k) = \frac{\sqrt{a}}{\sqrt{2\pi}} \frac{2a}{a^2 + k^2} \)

shows that the particle's momentum is likely to be around the zero value (i.e., centering near \( k = 0 \)). This distribution becomes more pronounced as \( a \) decreases, reflecting a more widespread initial wave function in position space.

The time-dependent Schrödinger equation is fundamental in quantum mechanics, describing how the quantum state of a physical system evolves over time. For our free particle wave function, we use this equation to compute \( \Psi(x, t) \), making it a function of time and position.
To construct \( \Psi(x, t) \), we integrate over all momentum space:

• \( \Psi(x, t) = \int_{-\infty}^{\infty} \phi(k) e^{i(kx - \frac{\hbar k^2}{2m} t)} \ dk \)

This transforms the Fourier coefficients back into real space, showing how the wave function modulates over time. Here, \( e^{-\frac{\hbar k^2}{2m} t} \) represents the quantum state's evolution in time, borrowing elements from the Hamiltonian operator in the Schrödinger equation.
This process not only delivers \( \Psi(x, t) \) but also highlights the intricate dynamics governed by the wavefunction's momentum components, made clearer by integrating across all \(k\).