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Chapter 39: Problem 34

Particle \(A\) is described by the wave function \(\psi(x, y, z)\) . Particle \(B\) is described by the wave function \(\psi(x, y, z) e^{i \phi}\) , where \(\phi\) is a real constant. How does the probability of finding particle \(A\) within a volume \(d V\) around a certain point in space compare with the probability of finding particle \(B\) within this same volume?

Short Answer

Expert verified
The probabilities are the same for both particles.

Step by step solution

01

Understand probability for particle A

For particle A, we are given the wave function \( \psi(x, y, z) \). The probability density is given by the squared magnitude of the wave function, \( |\psi(x, y, z)|^2 \). The probability of finding particle A within a volume \( dV \) is \(|\psi(x, y, z)|^2 dV \).
02

Evaluate probability for particle B

Particle B has the wave function \( \psi(x, y, z) e^{i \phi} \). The presence of the phase \( e^{i \phi} \) does not affect the magnitude of the wave function since \( |e^{i \phi}| = 1 \). Therefore, the probability density is \( |\psi(x, y, z) e^{i \phi}|^2 = |\psi(x, y, z)|^2 \). So, the probability of finding particle B within the same volume \( dV \) is also \( |\psi(x, y, z)|^2 dV \).
03

Compare probabilities of particles A and B

Both particles have the same probability densities, \( |\psi(x, y, z)|^2 \). Therefore, the probability of finding either particle A or particle B within the volume \( dV \) is the same. The phase factor \( e^{i \phi} \) does not change the probability density.

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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, the wave function is a fundamental concept that describes the quantum state of a particle or a system of particles. The wave function is typically denoted by the symbol \( \psi(x, y, z) \), and it is a complex-valued function of position. This means it has both real and imaginary parts. The wave function provides crucial information about the particle, including its behavior and how it evolves over time.
  • **Purpose**: Describes the state of a quantum system and allows us to make predictions about physical properties like position, momentum, and energy.
  • **Nature**: It is generally a complex function, meaning that it can be expressed in the form \( \psi = a + bi \), where \( i \) is the imaginary unit.
The actual physical significance of the wave function lies not in the function itself, but rather in what can be derived from it, specifically the probability density.
Probability Density
The probability density in quantum mechanics provides the likelihood of finding a particle in a given region of space. It is derived from the wave function, specifically through the operation of taking the squared magnitude of the wave function. For a wave function \( \psi(x, y, z) \), the probability density is given by \( |\psi(x, y, z)|^2 \). This is a crucial concept because it bridges the gap between the wave function and measurable quantities.
  • **Calculation**: To find the probability density, you simply calculate the square of the absolute value of the wave function, \( |\psi|^2 \). For a wave function without a phase, this would look like \( |\psi(x, y, z)|^2 \).
  • **Probability within Volume**: Multiplying the probability density by a small volume \( dV \), such as \( |\psi(x, y, z)|^2 dV \), gives the probability of finding the particle within that volume.
This explanation makes it clear why the probability of finding particle A is \( |\psi(x, y, z)|^2 dV \), which relies entirely on the wave function's squared magnitude.
Phase Factor
In quantum mechanics, the phase factor is an additional component that can be part of a wave function, expressed as \( e^{i \phi} \), where \( \phi \) is a real-valued constant. While it might seem that adding such a factor could alter properties of a particle, it turns out this phase factor does not affect the probability density.
  • **Nature of the Phase Factor**: It is a complex exponential where \( |e^{i \phi}| = 1 \). This means that it has a magnitude of just 1 and therefore does not change the overall magnitude of the wave function.
  • **Impact on Probability Density**: Because the magnitude of the phase factor is 1, when you compute the probability density, \( |\psi(x, y, z) e^{i \phi}|^2 \) simplifies to \( |\psi(x, y, z)|^2 \), indicating that the presence of a phase factor does not alter measurable probabilities.
Thus, as shown in the original exercise, particles A and B have the same probability of being found within any given volume, despite the additional phase factor in particle B's wave function.

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Most popular questions from this chapter

You want to study a biological spocimen by mcans of a wavelength of \(10.0 \mathrm{nm},\) and you have a choice of using electromagnetic waves or an electron microscope. (a) Calculate the ratio of the energy of a 10.0 -nm- wavelength photon to the kinetic energy of a 10.0 -nm-wavelength electron. (b) In view of your answer to part (a), which would be less damaging to the specimen you are studying; photons or electrons?

Proton Energy in a Nucleus. The radii of atomic nuclei are of the order of \(5.0 \times 10^{-15} \mathrm{m}\) . (a) Estimate the minimum uncertainty in the momentum of a proton if it is confined within a nucleus. (b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the relativistic relationship between energy and momentum, Fq. \((37.39),\) to obtain an estimate of the kinetic energy of a proton confined within a nucleus, (c) For a proton to remain bound within a nucleus, what must the magnitude of the (negative) potential energy for a proton be within the nucleus? Give your answer in \(\mathrm{oV}\) and in MoV. Compare to the potential energy for an electron in a hydrogen atom, which has a magnitude of a few tens of eV. (This shows why the interaction that binds the nucleus together is called the "strong nuclear force.")

(a) What is the de Broglie wavelength of an electron accelerated from rest through a potential increase of 125 \(\mathrm{V} ?\) (b) What is the de Broglie wavelength of an alpha particle \((q=+2 e,\) \(m=6.64 \times 10^{-27} \mathrm{kg} )\) accelerated from rest through a potential drop of 125 \(\mathrm{V} ?\)

Wavelength of a Bullet. Calculate the de Broglie wavelength of a \(5.00-\) g bullet that is moving at 340 \(\mathrm{m} / \mathrm{s}\) . Will the bullet exhibit wavelike properties?

Compute \(|\Psi|^{2}\) for \(\Psi=\psi\) sin \(\omega t,\) where \(\psi\) is time independent and \(\omega\) is a real constant. Is this a wave function for a stationary state? Why or why not?
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