Chapter 40: Problem 4
A particle is described by a wave function \(\psi(x) = Ae^{-\alpha x^2}\), 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
Step by step solution
Understand the Wave Function
Observation about the Width of the Wave Packet
Relate Width to Uncertainty in Position
Apply the Heisenberg Uncertainty Principle
Determine Effect on Particle's Uncertainty in Momentum
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Wave Function
- Amplitude (A): Determines the wave's height but not the importance of spread.
- Gaussian Shape: The hallmark of this specific wave function, which implies rapid decrease as you move from the center.
Heisenberg Uncertainty Principle
- Position Uncertainty (\Delta x): How unsure we are about the particle's location.
- Momentum Uncertainty (\Delta p): How unsure we are about the particle's momentum.
Momentum Uncertainty
- As \(\alpha\) gets larger, the particle's momentum uncertainty, \(\Delta p\), increases to maintain the principle balance.
- This explains why it's challenging to measure exact momentum if a position is accurately known.
Position Uncertainty
- Position uncertainty decreases as \(\alpha\) increases: the wave packet tightens.
- More \(\alpha\) makes the particle's possible location more precise.