10 - Physics 6210/Spring 2007/Lecture 10 Lecture 10...

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Physics 6210/Spring 2007/Lecture 10 Lecture 10 Relevant sections in text: § 1.7 Gaussian state Here we consider the important example of a Gaussian state for a particle moving in 1-d. Our treatment is virtually identical to that in the text, but this example is sufficiently instructive to give it again here. We define this state by giving its components in the position basis, i.e., its wave function: ψ ( x ) = h x | ψ i = ± 1 π 1 / 4 d ² exp ± ikx - x 2 2 d 2 ² . You can check as a good excercise that this wave function is normalized: 1 = h ψ | ψ i = Z -∞ dx h ψ | x ih x | ψ i = Z -∞ dx | ψ ( x ) | 2 . Roughly speaking, the wave function is oscillatory with wavelength 2 π k but sitting in a Gaussian envelope centered at the origin. The probability density for position is a Gaussian centered at the origin with width determined by d . Thus this state represents a particle “localized” near the origin within a statistical uncertainty specified by d . Let us make this more precise and exhibit some properties of this state. As an exercise you can check the following results. h X i = h ψ | X | ψ i = Z -∞ dx x | ψ ( x ) | 2 = 0 . so that the mean location of the particle is at the origin in this state. Next we have h X 2 i = h ψ | X 2 | ψ i = Z -∞ dx x 2 | ψ ( x ) | 2 = d 2 2 . Thus the dispersion in position is h Δ X 2 i = d 2 2 , i.e., d 2 is the standard deviation of the probability distribution for position. Next we have h P i = h ψ | P | ψ i Z -∞ dx ψ * ( x ) ¯ h i d dx ψ ( x ) = ¯ hk, telling us that, on the average, this state has the particle moving with momentum ¯ hk , and h P 2 i = h ψ | P 2 | ψ i Z -∞ dx ψ * ( x )( - ¯ h 2 ) d 2 dx 2 ψ ( x ) = ¯ h 2 2 d 2 + ¯ h 2 k 2 , 1
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Physics 6210/Spring 2007/Lecture 10 so that the momentum uncertainty is h Δ P 2 i = ¯ h 2 2 d 2 . Thus the momentum uncertainty varies reciprocally with d relative to the position uncer- tainty. The product of position and momentum uncertainties is as small as allowed by the uncertainty relation. One sometimes calls | ψ i a minimum uncertainty state . Because the Fourier transform of a Gaussian function is another Gaussian, it happens
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10 - Physics 6210/Spring 2007/Lecture 10 Lecture 10...

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