Uncertainty in position for eigenstates (Liboff 3.10)

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SUMMARY

The discussion focuses on proving that for the wave function ψ(x,t) = A exp((x-x0)² / (4a²)) * exp(ip0x)/ħ * exp(-iω0t), the uncertainty in position squared, (Δx)², equals a². Participants express difficulty in manipulating the equation and suggest that a clear understanding of the algebraic substitutions and the implications of the parameter 'a' on the wave function's shape is essential for consistency in conclusions. The problem is sourced from Liboff's fourth edition, emphasizing the need for precise mathematical handling of quantum states.

PREREQUISITES
  • Understanding of wave functions in quantum mechanics
  • Familiarity with the concept of uncertainty in position (Δx)
  • Knowledge of algebraic manipulation and substitution techniques
  • Basic principles of quantum mechanics as outlined in Liboff's textbook
NEXT STEPS
  • Review the derivation of uncertainty principles in quantum mechanics
  • Study the implications of parameter changes on wave function shapes
  • Learn about the mathematical properties of Gaussian functions in quantum states
  • Explore the concept of expectation values in quantum mechanics
USEFUL FOR

Students of quantum mechanics, particularly those tackling problems related to wave functions and uncertainty principles, as well as educators seeking to clarify these concepts in a classroom setting.

WyoChuck
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[STRIKE][/STRIKE]

Homework Statement



From Liboff edition 4:

For the state

ψ(x,t)=A exp(x-x0)2 / {4a2} * exp(ip0x)/hbar * exp (-iω0t)

show that (Δx)2 = a2

then argue the consistency of this conclusion with the change in shape that |ψ2| suffers with a change in the parameter a

Homework Equations



Equations given in problem as well as

(Δx)2 = <(x-<x>2)>


The Attempt at a Solution



So far I've tried substituting in Δx and using algebra to eliminate variables as well as solving for just a but unfortunately have been unable to make and leeway into proving the solution. In general I'm just lost as to where to start and any usefull advice will come in quite handy.
 
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WyoChuck said:
[STRIKE][/STRIKE]

Homework Statement



From Liboff edition 4:

For the state

ψ(x,t)=A exp(x-x0)2 / {4a2} * exp(ip0x)/hbar * exp (-iω0t)

show that (Δx)2 = a2

then argue the consistency of this conclusion with the change in shape that |ψ2| suffers with a change in the parameter a

Homework Equations



Equations given in problem as well as

(Δx)2 = <(x-<x>2)>


The Attempt at a Solution



So far I've tried substituting in Δx and using algebra to eliminate variables as well as solving for just a but unfortunately have been unable to make and leeway into proving the solution. In general I'm just lost as to where to start and any usefull advice will come in quite handy.
I have no idea what "substituting in Δx and using algebra to eliminate variables as well" means. Show your actual work.
 

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