Formal Developments
We have to go through a few more formalities before we can resume our discussion of quantum effects in physics. In particular, we need to address minimal uncertainties of observables in quantum mechanics, and we have to discuss transformation and solution
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Formal Developments
We have to go through a few more formalities before we can resume our discussion of quantum effects in physics. In particular, we need to address minimal uncertainties of observables in quantum mechanics, and we have to discuss transformation and solution properties of differential operators. I have also included an introduction to the notion of length dimensions of states, since this is useful for understanding the meaning of matrix elements in scattering theory in Chaps. 11 and 13. Furthermore, I have included a section on frequencytime Fourier transformation, although that can only be defined in a distributional sense for time-dependent wave functions. However, it is sometimes useful to represent the decompositions of states |ψ(t) in terms of energy eigenmodes |ψα , H |ψα = Eα |ψα , in the framework of Fourier transformation to a frequencydependent state |ψ(ω). The frequency-dependent states vanish if h¯ ω is not part of the spectrum of H , and they contain δ-functions for hω ¯ in the discrete spectrum of H .
5.1 Uncertainty Relations The statistical interpretation of the wave function naturally implies uncertainty in an observable Ao if the wave function is not an eigenstate of the hermitian operator A that corresponds to Ao . Suppose that A has eigenvalues an , A|φn = an |φn ,
φm |φn = δmn .
(5.1)
Substitution of the expansion |ψ =
|φn φn |ψ
(5.2)
n
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_5
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5 Formal Developments
into the formula for the expectation value A = ψ|A|ψ in the state |ψ yields A =
an |φn |ψ|2 ,
(5.3)
n
i.e. |φn |ψ|2 is a probability that the value an for the observable Ao will be observed if the system is in the state |ψ. If the distribution |φn |ψ|2 is strongly concentrated around a particular index , then it is very likely that a measurement of Ao will find the value a with very little uncertainty. However, if the probability distribution |φn |ψ|2 covers a broad range of indices or has maxima e.g. for two separated indices, then there will be high uncertainty of the value of the observable Ao , and observation of Ao for many copies of the system in the state |ψ will yield a large scatter of observed values. If A has a continuous spectrum of eigenvalues, e.g. if A = x is the operator for the location x of a particle in one dimension, then |x|ψ|2 dx is the probability to find the system with a value of x in the interval [x, x + dx]. Heisenberg found in 1927 an intuitive estimate for the minimal product of uncertainties x and p in location and momentum of a particle [75]. His arguments were easily made rigorous and generalized to other pairs of observables using the statistical formalism of quantum mechanics. Suppose that two observables Ao and Bo are represented by the two hermitian operators A and B. The expectation value of the observ
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