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Chapter 39: Problem 29

Consider a wave function given by \(\psi(x)=A \sin k x,\) where \(k=2 \pi / \lambda\) and \(A\) is a real constant. (a) For what values of \(x\) is there the highest probability of finding the particle described by this wave function? Explain. (b) For which values of \(x\) is the probability zero? Explain.

Short Answer

Expert verified
(a) Highest probability at \( x = \frac{\lambda}{4} + n\frac{\lambda}{2} \). (b) Zero probability at \( x = \frac{n\lambda}{2} \).

Step by step solution

01

Understanding the Wave Function

The wave function is given by \( \psi(x) = A \sin(kx) \). Here, \( A \) is a constant amplitude, and \( \sin(kx) \) is the sine function that describes the variation of the wave along the x-axis. The probability density is related to the magnitude square of the wave function.
02

Expressing Probability Density

The probability density for a wave function \( \psi(x) \) is given by \( \vert \psi(x) \vert^2 \). For our function, this becomes \( \vert A \sin(kx) \vert^2 = A^2 \sin^2(kx) \). The probability density peaks when \( \sin^2(kx) \) is maximized.
03

Finding Maximum Probability Density

The sine squared function \( \sin^2(kx) \) reaches its maximum value of 1 when \( \sin(kx) = \pm 1 \). This occurs when \( kx = \frac{\pi}{2} + n\pi \), \( n \in \mathbb{Z} \). Thus, the values of \( x \) where the probability is maximized are \( x = \frac{\pi}{2k} + \frac{n\pi}{k} \). Substituting \( k = \frac{2 \pi}{\lambda} \), we have \( x = \frac{\lambda}{4} + n\frac{\lambda}{2} \).
04

Determining Zero Probability Values

The probability density is zero when \( \sin^2(kx) = 0 \), or equivalently, when \( \sin(kx) = 0 \). This happens at \( kx = n\pi \), \( n \in \mathbb{Z} \). Hence, the values of \( x \) that yield zero probability are \( x = \frac{n\lambda}{2} \), thus corresponding to the nodes of the wave.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Function
In quantum mechanics, a wave function, often denoted as \( \psi(x) \), provides a full description of a quantum state. It relates to the probability of finding a particle in a particular position. In the exercise, the wave function is described by \( \psi(x) = A \sin(kx) \), where \( A \) is a constant amplitude that affects the overall scale of the wave. The sine function here introduces periodicity, meaning the wave repeats itself at regular intervals.
Understanding the wave function is crucial because it not only explains where a particle is likely to be found but also how it behaves in space. This function is central to solving various problems in quantum mechanics, including energy levels and probability calculations.
Remember, the wave function is a key to visualizing a particle's behavior, even if it's not directly observable.
Probability Density
The probability density is a crucial concept, denoted by the square of the wave function's magnitude: \( |\psi(x)|^2 \). It gives the likelihood of finding a particle at a position \( x \). For the sine wave function in the given exercise, this translates to \( A^2 \sin^2(kx) \). This expression shows how the probability distribution changes with x.
Points where \( \sin^2(kx) \) reaches its maximum, specifically when \( \sin(kx) = \pm 1 \), correspond to the highest probability densities. For these maxima, the probability density reaches \( A^2 \). Therefore, peaks in probability density occur at specific positions within the wave, which are crucial in predicting particle behavior.
Sine Function
The sine function, \( \sin(x) \), is periodic and oscillatory, characterized by its peaks and troughs. In the context of our wave function, \( \sin(kx) \) determines the framework of the wave's periodic behavior. The variable \( k \) is the wave number, linked to the wave's wavelength by \( k = \frac{2\pi}{\lambda} \).
  • Maximum Values: \( \sin(kx) = \pm 1 \) occurs at \( kx = \frac{\pi}{2} + n\pi \), leading to maximal probability densities.
  • Zeros: When \( \sin(kx) = 0 \), corresponding probability densities are zero, which occur at the nodes.
These behaviors underpin the periodic and structural aspects of the quantum wave, crucial for understanding quantum phenomena.
Wave Nodes
Wave nodes are the points along a wave where the amplitude is consistently zero. For our wave function, this means the probability density is also zero at these points. They're determined by when \( \sin(kx) = 0 \), which is at \( kx = n\pi \) for \( n \in \mathbb{Z} \). In terms of \( x \), nodes appear at \( x = \frac{n\lambda}{2} \).
These nodes have significant implications in quantum mechanics because they indicate specific positions where a particle is never found. Understanding where nodes occur can help predict the likelihood of particle presence and the wave's overall pattern.

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Most popular questions from this chapter

High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression \(\lambda=h / p\) still holds, but we must use the rclativistic cxpression for momcntum, \(p=m v / \sqrt{1-v^{2} / c^{2}}\) (a) Show that the speed of an electron that has de Broglie wavelength \(\lambda\) is $$ v=\frac{c}{\sqrt{1+(m c \lambda / h)^{2}}} $$ (b) The quantity \(h / m c\) equals \(2426 \times 10^{-12} \mathrm{m}\) (As we saw in Section 38.7 , this same quantity appears in Eq. (38.23), the expression for Compton scattering of photons by electrons) If \(\lambda\) is small compared to \(h / m c,\) the denominator in the expression found in part (a) is close to unity and the speed \(v\) is very close to \(c\) . In this case it is convenient to write \(v=(1-\Delta) c\) and express the speed of the electron in terms of \(\Delta\) rather than \(v\) . Find an expression for \(\Delta\) valid when \(\lambda \ll h / m c\) . [Hine. Use the binomial expansion \((1+z)^{n}=1+n z+n(n-1) z^{2} / 2+\cdots,\) valid for the case \(|z| < 1 .\) (c) How fast must an electron move for its de Broglie wavelength to be \(1.00 \times 10^{-15} \mathrm{m}\) , comparable to the size of a proton? Express your answer in the form \(v=(1-\Delta) c,\) and state the value of \(\Delta .\)

A particle is described by the normalized wave function \(\psi(x, y, z)=A x e^{-\alpha x^{2}} e^{-\beta \beta} e^{-\gamma x^{2}},\) where \(A, \alpha, \beta,\) and \(\gamma\) are all real, positive constants. The probability that the particle will be found in the infinitesimal volume \(d x d y d z\) centered at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) is \(\left|\psi\left(x_{0}, y_{0}, z_{0}\right)\right|^{2} d x d y d z\) (a) At what value of \(x_{0}\) is the particle most likely to be found? (b) Are there values of \(x_{0}\) for which the probability of the particle being found is zero? If so, at what \(x_{0} ?\)

Why Don't We Diffract? (a) Calculate the de Broglie wavelength of a typical person walking through a doorway. Make reasonable approximations for the necessary quantitics. (b) Will the person in part (a) exhibit wave-like behavior when walking through the "single slit" of a doorway? Why?

(a) Approximately how fast should an electron move so it has a wavelength that makes it useful to measure the distance between adjacent atoms in typical crystals (about 0.10 \(\mathrm{nm} ) ?\) (b) What is the kinetic energy of the electron in part (a)?(c) What would be the energy of a photon of the same wavelength as the electron in part \((b) ?(d)\) Which would make a more effective probe of small-scale structures, electrons or photons? Why?

A 10.0 -g marble is gently placed on a horizontal tabletop that is 1.75 \(\mathrm{m}\) wide. (a) What is the maximum uncertainty in the horizontal position of the marble? (b) According to the Heisenberg uncertainty principle. what is the minimum uncertainty in the horizontal velocity of the marble? (c) In light of your answer to part (b), what is the longest time the marble could remain on the table? Compare this time to the age of the universe, which is approximately 14 billion years. (Hint Can you know that the horizontal velocity of the marble is exactly zero?)
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