Lect16_Wavepackets

# Lect16_Wavepackets - Global Phase of a Pure State Statement...

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Lecture 16: Wavepackets PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Global Phase of a Pure State Statement: True for any ! , | "# Analogous to a gauge invariance You are free to choose ! which is most convenient for a particular problem Proof: All measurable quantities are of the form \$" | A | "# let | "# ! e i ! | "# : Does ! " = i e A A e e Ae e A i i i i = = # \$ \$ ? 2 1 2 1 2 1 + = + i i e e

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Relative Phase in a Superposition Answer: No, this is an incorrect application of the phase invariance principle The statement doesn’t apply separately to individual terms in a superposition state The correct application is: Global phase is always undetectable Only relative phase between terms in a superposition can be measured Such as ! 2 % 1 above For example: e i " 1 # 1 + e i 2 2 = e i 1 1 + e i 2 \$ 1 ( ) 2 ( ) = 1 + e i 2 \$ 1 ( ) 2 1 1 " 2 ( ) = 1 2 " 1 ( ) Interference Fringes Let The probability density is: Location of fringes depends on relative phase, so can be observed A device to measure the relative phase is called an interferometer = 1 k + e i \$ k ( ) = 1 e i / 2 e \$ i / 2 k + e i / 2 \$ k ( ) P ( x ) = ( x ) 2 = 2cos 2 kx # \$ /2 ( ) = 1 + cos 2 kx # ( ) Beam of particles 50/50 beam splitter mirror Detector fringes
Free Particle Eigenstates: Consider a free particle in an energy/momentum eigenstate: How does it evolve in time? Only the global phase changes Global phase is not measureable Is the particle really moving? m P H 2 2 = 0 2 2 0 ) ( k e t m t k i h ! = " 0 ) 0 ( k = ! Wavefunction in K-space What is the wavefunction in K-representation?

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Lect16_Wavepackets - Global Phase of a Pure State Statement...

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