Question

Two wave functions are distinguishable if they lead to a different probability density. Which of the following wave functions are distinguishable from \(\sin k x ?\) a. \(\left(e^{i k x}-e^{-i k x}\right) / 2\) b. \(e^{i \theta} \sin k x, \theta\) a constant c. \(\cos (k x-\pi / 2)\) d. \(i \cos (k x+\pi / 2)(\sin \theta+i \cos \theta)\left(-\frac{\sqrt{2}}{2}+i \frac{\sqrt{2}}{2}\right)\) \(\theta\) is a constant

Short answer

Upon comparing the probability densities, we find that wave functions a and b are indistinguishable from \(\sin kx\) as their probabilities densities are the same as that of \(\sin kx\): \(P_a(x) = P_b(x) = P_{ref}(x) = \sin^2 kx\). Wave functions c and d have different probability densities than \(\sin kx\), hence they are distinguishable from \(\sin kx\): \(P_c(x) = \sin^2(kx)\), and \(P_d(x)\) also results in a different probability density. So, wave functions c and d are distinguishable from \(\sin kx\).

Step-by-step solution

Calculate the probability density \(P_{ref}(x)\) of the reference wave function \(\psi_{ref}(x) =\sin kx\) as follows: \(P_{ref}(x) = |\sin kx|^2 = \sin^2 kx\) #Step 2: Analyze wave function a#

First, calculate the probability density \(P_a(x)\) of the wave function \(a\): \(\psi_a(x) = \frac{(e^{ikx} - e^{-ikx})}{2}\). \(P_a(x) = \left|\frac{(e^{ikx} - e^{-ikx})}{2}\right|^2 = \frac{1}{4}|(e^{ikx} - e^{-ikx})|^2\) Recall the Euler's formula: \(e^{ix} = \cos x + i\sin x\) Using the Euler's formula, evaluate \(P_a(x)\) to compare with \(P_{ref}(x)\). #Step 3: Analyze wave function b#

Similarly, calculate the probability density \(P_b(x)\) of the wave function \(b\): \(\psi_b(x) = e^{i\theta}\sin kx\), with \(\theta\) a constant. \(P_b(x) = |e^{i\theta}\sin kx|^2\) Evaluate \(P_b(x)\) and compare with \(P_{ref}(x)\). #Step 4: Analyze wave function c#

Calculate the probability density \(P_c(x)\) of the wave function \(c\): \(\psi_c(x) = \cos(kx - \frac{\pi}{2})\). \(P_c(x) = |\cos(kx - \frac{\pi}{2})|^2\) Evaluate \(P_c(x)\) and compare with \(P_{ref}(x)\). #Step 5: Analyze wave function d#

Calculate the probability density \(P_d(x)\) of the wave function \(d\): \(\psi_d(x) = i\cos(kx + \frac{\pi}{2})(\sin\theta + i\cos\theta)\left(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right)\), with \(\theta\) a constant. \(P_d(x) = \left|i\cos(kx + \frac{\pi}{2})(\sin\theta + i\cos\theta)\left(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right)\right|^2\) Evaluate \(P_d(x)\) and compare with \(P_{ref}(x)\). #Step 6: Determine which wave functions are distinguishable from \(\sin kx\) #

Determine the distinguishability of each wave function based on their probability density comparisons with \(P_{ref}(x)\). If the probability densities are different, the wave functions are distinguishable.

Key concepts

Probability Density

Probability density is a core concept in quantum mechanics that helps us understand where a particle is likely to be found at a certain point in time and space. It is calculated by taking the square of the magnitude of a wave function. For example, for the wave function \(\psi(x) = \sin kx\), the probability density is given by \(P(x) = |\sin kx|^2 = \sin^2 kx\).

Calculating probability density helps us visualize the position likelihood of the particle. It provides a critical link between the abstract mathematics of wave functions and the physical reality of particle locations. In the context of our exercise, comparing the probability densities of various wave functions allows us to determine if they are distinguishable from the reference wave function.

Euler's Formula

Euler's Formula is a fundamental mathematical expression that links complex exponentials to trigonometric functions. It is given by \(e^{ix} = \cos x + i\sin x\).

This formula is particularly useful in the study of wave functions, as it helps transform exponential expressions into trigonometric forms. For instance, in analyzing the probability density of wave function \(a\), \(\psi_a(x) = \frac{(e^{ikx} - e^{-ikx})}{2}\), Euler's formula aids in converting exponential terms to sine and cosine functions to simplify calculations.

By using Euler's formula, complex wave functions can be understood through more intuitive trigonometric functions, allowing a seamless transition between the different representations and simplifying comparisons between probabilities.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate different trigonometric functions to one another. These identities are pivotal for simplifying wave functions and probability calculations. For instance, knowing that \(\cos(kx - \frac{\pi}{2}) = \sin kx\) immediately allows us to realize that the probability density calculated from this is the same as the reference wave function \(\psi_{ref}(x) = \sin kx\).

Utilizing trigonometric identities effectively reduces the complexity in quantum mechanics problems involving wave functions. It transforms solutions into simpler forms and reveals hidden correspondences between different functions, helping us identify when two wave functions are indistinguishable because they have the same probability density.

Quantum Mechanics

Quantum mechanics is the branch of physics dealing with phenomena at the smallest scales of energy levels of atoms and subatomic particles. One of its foundational concepts is the wave function, which gives a probabilistic description of a particle's state, encapsulating all probabilities of a system's measurable properties.

In this exercise, wave functions represent the state of a particle in a space, and their distinguishability is determined by comparing probability densities. Quantum mechanics principles are what make some aspects of the exercise so abstract and mathematically intense.

Understanding these principles sheds light on the behavior of particles at quantum scales, emphasizing why wave functions provide only probabilistic, rather than certain, predictions of a particle's position or momentum. This probabilistic nature is one of the defining characteristics that sets quantum mechanics apart from classical physics.