2. (a) What are incompatible observables? (b) The generalised uncertainty principle for the operators  and  is: 2 10 ≥ (A, B)))² Explain the terms in this equation and state how the quantity σ is deter- mined for the operator A. Explain the relevance of the equation given to compatible/incompatible observables. (c) What property of Hermitian operators ensures that the right side of the equation given in part (b) is real if A and B are Hermitian? (d) Why is the energy-time uncertainty principle fundamentally different from the generalised uncertainty principle given in (b)? (e) Use the energy-time uncertainty principle to estimate the rest-mass energy of the pion. Assume that the range of the strong nuclear force, for which the pion is the carrier, is approximately 1 × 10-15 m and that the pion travels at the speed of light. What is the corresponding mass of the pion? (f) What can you say about the lifetime of an atomic state, if the photon that it emits when it decays has a narrow spectral linewidth?
2. (a) What are incompatible observables? (b) The generalised uncertainty principle for the operators  and  is: 2 10 ≥ (A, B)))² Explain the terms in this equation and state how the quantity σ is deter- mined for the operator A. Explain the relevance of the equation given to compatible/incompatible observables. (c) What property of Hermitian operators ensures that the right side of the equation given in part (b) is real if A and B are Hermitian? (d) Why is the energy-time uncertainty principle fundamentally different from the generalised uncertainty principle given in (b)? (e) Use the energy-time uncertainty principle to estimate the rest-mass energy of the pion. Assume that the range of the strong nuclear force, for which the pion is the carrier, is approximately 1 × 10-15 m and that the pion travels at the speed of light. What is the corresponding mass of the pion? (f) What can you say about the lifetime of an atomic state, if the photon that it emits when it decays has a narrow spectral linewidth?
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Transcribed Image Text:2. (a) What are incompatible observables?
(b) The generalised uncertainty principle for the operators  and  is:
2
10 ≥ (A, B)))²
Explain the terms in this equation and state how the quantity σ is deter-
mined for the operator A. Explain the relevance of the equation given to
compatible/incompatible observables.
(c) What property of Hermitian operators ensures that the right side of the
equation given in part (b) is real if A and B are Hermitian?
(d) Why is the energy-time uncertainty principle fundamentally different from
the generalised uncertainty principle given in (b)?
(e) Use the energy-time uncertainty principle to estimate the rest-mass energy
of the pion. Assume that the range of the strong nuclear force, for which
the pion is the carrier, is approximately 1 × 10-15 m and that the pion
travels at the speed of light. What is the corresponding mass of the pion?
(f) What can you say about the lifetime of an atomic state, if the photon that
it emits when it decays has a narrow spectral linewidth?
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