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Chapter 40: Problem 8

A particle is described by a wave function \(\psi(x)=A e^{-\alpha x^{2}}\) where \(A\) and \(\alpha\) are real, positive constants. If the value of \(\alpha\) is increased, what effect does this have on (a) the particle's uncertainty in position and (b) the particle's uncertainty in momentum? Explain your answers.

Short Answer

Expert verified
Increasing \(\alpha\) decreases position uncertainty \(\Delta x\), but increases momentum uncertainty \(\Delta p\).

Step by step solution

01

Understand the wave function

The given wave function is \(\psi(x)=A e^{-\alpha x^{2}}\). This is a Gaussian wave packet centered at \(x=0\) with a spread influenced by \(\alpha\). A greater \(\alpha\) implies a narrower spread in the position space.
02

Uncertainty in position

The uncertainty in position \(\Delta x\) for a Gaussian wave packet is given by \(\Delta x = \frac{1}{\sqrt{2\alpha}}\). This means that as \(\alpha\) increases, \(\Delta x\) decreases, indicating a smaller uncertainty in position.
03

Uncertainty in momentum

According to the Heisenberg Uncertainty Principle, \(\Delta x \Delta p \geq \frac{\hbar}{2}\). Since \(\Delta x\) decreases as \(\alpha\) increases, \(\Delta p\), the uncertainty in momentum, must correspondingly increase to satisfy the uncertainty principle.
04

Dual implications of increasing \(\alpha\)

Therefore, increasing \(\alpha\) results in a more precise position measurement (smaller \(\Delta x\)), but a larger uncertainty in momentum (larger \(\Delta p\)). This is a demonstration of the inverse relationship between position and momentum uncertainties.

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Key Concepts

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

Gaussian wave packet
A Gaussian wave packet is a specific type of wave function that resembles a bell curve. It is highly symmetrical and smooth, representing a localized particle effectively. The mathematical expression for a Gaussian wave packet is often in the form \(\psi(x) = A e^{- \alpha x^2}\), where \(A\) is a normalization constant, and \(\alpha\) determines the spread of the wave packet.
The spread of the wave packet tells us how confined or spread out the particle is in space.
  • If \(\alpha\) is small, the wave packet is widely spread, indicating a large degree of uncertainty about where the particle is.
  • If \(\alpha\) is large, the wave packet becomes narrower, meaning the particle's position is more confined, and we have greater certainty about its location.
This characteristic shape and behavior make Gaussian wave packets useful for studying uncertainty in quantum mechanics.
Wave function
A wave function is a fundamental concept in quantum mechanics, describing the quantum state of a particle. It provides a complete description of all possible configurations of a system.
In mathematical terms, it's a complex-valued function, typically denoted as \(\psi(x)\), that gives the probability amplitude of a particle's position in one dimension.
The square of the wave function's magnitude, \(|\psi(x)|^2\), gives the probability density of finding the particle at a specific position.
  • A wave function's form, as seen with \(\psi(x) = A e^{-\alpha x^2}\), can convey important information about the particle's motion and behavior.
  • Such functions help predict outcomes and properties of quantum systems, allowing calculations of various uncertainties like position and momentum.
Understanding wave functions is key to grasping concepts like the Heisenberg Uncertainty Principle.
Position uncertainty
Position uncertainty refers to the lack of precise knowledge about where a particle is located within a certain region. In quantum mechanics, this is inherently tied to the nature of wave functions.
For a Gaussian wave packet, the position uncertainty \(\Delta x\) is mathematically represented by \(\Delta x = \frac{1}{\sqrt{2\alpha}}\).
This indicates that if the parameter \(\alpha\) increases, then the position uncertainty decreases, resulting in a more precisely known position.
  • The more localized a wave function is (narrower wave packet), the smaller the position uncertainty.
  • The trend of \(\Delta x\) decreasing with increasing \(\alpha\) is an expression of this precision-enlargement relationship.
Position uncertainty is a crucial concept as it reflects the probabilistic nature of quantum mechanics.
Momentum uncertainty
Momentum uncertainty is the measure of how much we don't know about a particle's momentum, and it's reciprocally related to position uncertainty. According to the Heisenberg Uncertainty Principle, this relationship is given by \(\Delta x \Delta p \geq \frac{\hbar}{2}\), where \(\Delta p\) is the momentum uncertainty.
When position uncertainty \(\Delta x\) decreases with an increase in \(\alpha\), \(\Delta p\) must increase to satisfy the equation. This trade-off is essential in quantum mechanics.
  • A lesser spread in position implies a higher uncertainty in momentum and vice versa, demonstrating the balance captured by the uncertainty principle.
  • Understanding this balance showcases the limits of what can be simultaneously known about a particle's position and momentum.
Momentum uncertainty is a fundamental aspect of quantum systems and reflects the inherent limitations posed by quantum theory.

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Most popular questions from this chapter

Photon in a Dye Laser. An electron in a long, organic molecule used in a dye laser behaves approximately like a particle in a box with width 4.18 nm. What is the wavelength of the photon emitted when the electron undergoes a transition (a) from the first excited level to the ground level and (b) from the second excited level to the first excited level?

Protons, neutrons, and many other particles are made of more fundamental particles called quarks and antiquarks (the antimatter equivalent of quarks). A quark and an antiquark can form a bound state with a variety of different energy levels, each of which corresponds to a different particle observed in the laboratory. As an example, the \(\psi\) particle is a low-energy bound state of a so-called charm quark and its antiquark, with a rest energy of 3097 MeV; the \(\psi(2 S)\) particle is an excited state of this same quark-antiquark combination, with a rest energy of 3686 MeV. A simplified representation of the potential energy of interaction between a quark and an antiquark is \(U(x)=A|x|,\) where \(A\) is a positive constant and \(x\) represents the distance between the quark and the antiquark. You can use the WKB approximation (see Challenge Problem 40.72 ) to determine the bound-state energy levels for this potential-energy function. In the WKB approximation, the energy levels are the solutions to the equation $$\int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2} \quad(n=1,2,3, \ldots)$$ Here \(E\) is the energy, \(U(x)\) is the potential-energy function, and \(x=a\) and \(x=b\) are the classical turning points (the points at which \(E\) is equal to the potential energy, so the Newtonian kinetic energy would be zero). (a) Determine the classical turning points for the potential \(U(x)=A|x|\) and for an energy \(E\) . (b) Carry out the above integral and show that the allowed energy levels in the WKB approximation are given by $$E_{n}=\frac{1}{2 m}\left(\frac{3 m A h}{4}\right)^{2 / 3} n^{2 / 3} \quad(n=1,2,3, \ldots)$$ (Hint: The integrand is even, so the integral from \(-x\) to \(x\) is equal to twice the integral from 0 to \(x\) . \((\mathrm{c})\) Does the difference in energy between successive levels increase, decrease, or remain the same as \(n\) increases? How does this compare to the behavior of the energy levels for the harmonic oscillator? For the particle in a box? Can you suggest a simple rule that relates the difference in energy between successive levels to the shape of the potential-energy function?

The penetration distance \(\eta\) in a finite potential well is the distance at which the wave function has decreased to 1\(/ e\) of the wave function at the classical turning point: $$\psi(x=L+\eta)=\frac{1}{e} \psi(L)$$ The penetration distance can be shown to be $$\eta=\frac{\hbar}{\sqrt{2 m\left(U_{0}-E\right)}}$$ The probability of finding the particle beyond the penetration distance is nearly zero. (a) Find \(\eta\) for an electron having a kinetic energy of 13 eV in a potential well with \(U_{0}=20 \mathrm{eV} .\) (b) Find \(\eta\) for a 20.0 -MeV proton trapped in a 30.0 -Me \(\mathrm{V}\) -deep potential well.

An electron with initial kinetic energy 6.0 \(\mathrm{eV}\) encounters a barrier with height 11.0 \(\mathrm{eV}\) . What is the probability of tunneling if the width of the barrier is (a) 0.80 \(\mathrm{nm}\) and (b) 0.40 \(\mathrm{nm} ?\)

An electron is moving as a free particle in the \(-x\) -direction with momentum that has magnitude \(4.50 \times 10^{-24} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} .\) What is the one-dimensional time-dependent wave function of the electron?
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