Problem 10 - University Physics with Modern Physics

Question

A particle moving in one dimension (the \(x\) -axis) is described by the wave function \(\psi(x)=\left\\{\begin{array}{ll}{A e^{-b x},} & {\text { for } x \geq 0} \\ {A e^{b x},} & {\text { for } x<0}\end{array}\right.\) where \(b=2.00 \mathrm{m}^{-1}, A>0,\) and the \(+x\) -axis points toward the right. (a) Determine \(A\) so that the wave function is normalized. (b) Sketch the graph of the wave function. (c) Find the probability of finding this particle in each of the following regions: (i) within 50.0 \(\mathrm{cm}\) of the origin, (ii) on the left side of the origin (can you first guess the answer by looking at the graph of the wave function? (iii) between \(x=0.500 \mathrm{m}\) and \(x=1.00 \mathrm{m} .\)

Step-by-Step Solution

Answer

a) \(A = \sqrt{2} \, \mathrm{m}^{-\frac{1}{2}}\); b) Sketch shows exponential growth and decay; c) i) 0.432, ii) 0.5, iii) 0.017.

Step 1: Understand the Problem

We need to normalize the wave function \( \psi(x) \), graph it, and then find the probability of finding the particle in specified regions along the \( x \)-axis.

Step 2: Normalization Condition

To normalize the wave function, the integral of \( |\psi(x)|^2 \) over all space must equal 1:\[ \int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 1 \]Split the integral into two parts for \( x < 0 \) and \( x \geq 0 \): \[ \int_{-\infty}^0 |A e^{b x}|^2 \, dx + \int_0^{\infty} |A e^{-b x}|^2 \, dx = 1 \]

Step 3: Calculate Each Integral

For \( x < 0 \), calculate:\[ \int_{-\infty}^0 (A e^{b x})^2 \, dx = A^2 \int_{-\infty}^0 e^{2b x} \, dx = A^2 \left[ \frac{e^{2b x}}{2b} \right]_{-\infty}^0 = \frac{A^2}{2b} \]For \( x \geq 0 \), calculate:\[ \int_0^{\infty} (A e^{-b x})^2 \, dx = A^2 \int_0^{\infty} e^{-2b x} \, dx = A^2 \left[ -\frac{e^{-2b x}}{2b} \right]_0^{\infty} = \frac{A^2}{2b} \]

Step 4: Solve for Normalization Constant \( A \)

Adding both parts from Step 3 gives:\[ \frac{A^2}{2b} + \frac{A^2}{2b} = 1 \]\[ \frac{A^2}{b} = 1 \]\[ A^2 = b \]\[ A = \sqrt{b} \]Since \( b = 2.00 \, \mathrm{m}^{-1} \), then \( A = \sqrt{2.00} = \sqrt{2} \, \mathrm{m}^{-\frac{1}{2}} \).

Step 5: Sketch the Graph of the Wave Function

Plot the function \( \psi(x) \): it decays exponentially with \( e^{-2x} \) for \( x \geq 0 \) and grows with \( e^{2x} \) for \( x < 0 \). The function is symmetric in terms of its exponential decay characteristics about \( x = 0 \).

Step 6: Probability (i) within 50.0 cm of the Origin

Convert \(50.0 \, \mathrm{cm} = 0.50 \, \mathrm{m}\).Calculate:\[ \int_{-0.50}^{0.50} |\psi(x)|^2 \, dx = \int_{-0.50}^{0} (A e^{2x})^2 \, dx + \int_{0}^{0.50} (A e^{-2x})^2 \, dx \] Evaluating these integrals:\[ \int_{-0.50}^{0} e^{4x} \, dx = \left[ \frac{e^{4x}}{4} \right]_{-0.50}^{0} = \frac{1 - e^{-2}}{4} \]\[ \int_{0}^{0.50} e^{-4x} \, dx = \left[ -\frac{e^{-4x}}{4} \right]_{0}^{0.50} = \frac{1 - e^{-2}}{4} \]Thus, total probability is \( \frac{1 - e^{-2}}{2} \approx 0.432 \).

Step 7: Probability (ii) on the Left Side of the Origin

Calculate:\[ \int_{-\infty}^{0} |\psi(x)|^2 \, dx = \frac{A^2}{2b} = \frac{1}{2} \]This represents the probability of 0.5 because of the symmetric nature of the exponential function around the origin.

Step 8: Probability (iii) between 0.500 m and 1.00 m

Calculate:\[ \int_{0.50}^{1.00} |\psi(x)|^2 \, dx = A^2 \int_{0.50}^{1.00} e^{-4x} \, dx \]Evaluating gives:\[ \left[ -\frac{e^{-4x}}{4} \right]_{0.50}^{1.00} = \frac{e^{-2} - e^{-4}}{4} \approx 0.017 \].

Key Concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes nature at the smallest scales of energy levels of atoms and subatomic particles. It challenges our classical intuition with concepts like superposition and entanglement. In this context, the wave function, often denoted as \( \psi(x) \), is a central concept. This function provides a probabilistic description of a particle's quantum state as it exists in all possible areas. It is a complex-valued function, and its absolute square, \( |\psi(x)|^2 \), gives the probability density of a particle's position concerning the x-axis.
Thus, the wave function must be normalized, meaning the total probability of finding a particle anywhere on the x-axis must equal one. To achieve this, the integral of \( |\psi(x)|^2 \) over the whole space should sum up to one. These fundamental principles are vital as they transition concepts from classical probability to quantum probabilities, where certainty in position vanishes, replaced by likelihoods.

Probability Calculations

In the realm of quantum mechanics, probability calculations involve integrating the squared amplitude of the wave function over a specified region. This task tells us the likelihood of finding a particle within a given interval. For instance, if we have a particle described by a wave function \( \psi(x) \), to find the probability that it exists within a range, say from \( a \) to \( b \), we compute:
\[ P(a \leq x \leq b) = \int_a^b |\psi(x)|^2 \, dx \]The normalization condition guarantees that the total probability across all such ranges sums to one. For practical examples:

• Integrating over a finite region gives the fraction of probability mass within that region.
• Understanding these calculations helps in making predictions about particle positions and behaviors that can be verified experimentally.

By using these calculations, physicists can ascertain more about the possible location of particles, testing them against experimental results to validate current theories.

Exponential Functions

Exponential functions frequently appear in quantum mechanics, especially in the representation of wave functions. They inherently describe growth and decay processes, which makes them essential for modeling the behavior of particles in bound and unbound states.For a wave function that involves exponentials, like \( \psi(x) = A e^{-bx} \) for \( x \ge 0 \), we observe certain properties:

• Exponential decay shows how probabilities diminish or decay from a certain point, aligning with how quantum particles are more likely near specific regions, like potential wells.
• For \( x < 0 \), where \( \psi(x) = A e^{bx} \) indicates growth - although such a term is less common in physically observable states, it's used here to illustrate mathematical principles.

These functions, with their consistent rates of change, serve as natural solutions to differential equations in quantum scenarios, describing phenomena such as radioactive decay or electronic behavior in fields. When graphed, these curves provide a visual representation of probability distribution, essential in grasping quantum states' locality.