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I 5. Suppose that a wavefunction is given by. 0 v (e,t) = { = { t < 0 expli(kx- (ER- iT/2)t/h)] t≥0' where ER and I> 0 are parameters. (a) What is y(x, t)|2 for t≥ 0? Note that this wavefunction could represent that of state (or particle) produced at time t = 0 which decays with a mean lifetime of T = ħ/T. (b) This wavefunction can be transformed the the energy domain via a Fourier Transform: Þ(x, E) = What is Þ(x, E) ? 1 √2V(x, t) expli(Et/h)] dt. 2π

I 5. Suppose that a wavefunction is given by. 0 v (e,t) = { = { t < 0 expli(kx- (ER- iT/2)t/h)] t≥0' where ER and I> 0 are parameters. (a) What is y(x, t)|2 for t≥ 0? Note that this wavefunction could represent that of state (or particle) produced at time t = 0 which decays with a mean lifetime of T = ħ/T. (b) This wavefunction can be transformed the the energy domain via a Fourier Transform: Þ(x, E) = What is Þ(x, E) ? 1 √2V(x, t) expli(Et/h)] dt. 2π

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I need help with question 5 part a and b.
I 5. Suppose that a wavefunction is given by. 0 t<0 v(x,t) = { expli(ka - (En-iT/2)t/h)] +20 where ER and I >0 are parameters. (a) What is y(x, t) |2 for t≥ 0? Note that this wavefunction could represent that of state (or particle) produced at time t = 0 which decays with a mean lifetime of T = ħ/T. (b) This wavefunction can be transformed the the energy domain via a Fourier Transform: Þ(x, E) = 1 -∞ (x, t) expli(Et/h)] dt. What is P(x, E) ? (c) What is P(x, E)|2 ? Plot it. The parameter I is called the Full Width at Half Maximum (FWHM). Does this make sense? Note that I'r = ħ is a particular case of the uncertainty principle that is an exact relation. It holds for states which decay according the exponential decay law (which is often the case, due to decay rates being constant). It implies that unstable states (or particles) necessarily have a finite energy width. Q Search
Transcribed Image Text:I 5. Suppose that a wavefunction is given by. 0 t<0 v(x,t) = { expli(ka - (En-iT/2)t/h)] +20 where ER and I >0 are parameters. (a) What is y(x, t) |2 for t≥ 0? Note that this wavefunction could represent that of state (or particle) produced at time t = 0 which decays with a mean lifetime of T = ħ/T. (b) This wavefunction can be transformed the the energy domain via a Fourier Transform: Þ(x, E) = 1 -∞ (x, t) expli(Et/h)] dt. What is P(x, E) ? (c) What is P(x, E)|2 ? Plot it. The parameter I is called the Full Width at Half Maximum (FWHM). Does this make sense? Note that I'r = ħ is a particular case of the uncertainty principle that is an exact relation. It holds for states which decay according the exponential decay law (which is often the case, due to decay rates being constant). It implies that unstable states (or particles) necessarily have a finite energy width. Q Search
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straight psi left parenthesis straight x comma straight t right parenthesis equals 0 space space space for space straight t less than 0 space space space space space space space space space space equals straight e to the power of straight i left square bracket kx minus left parenthesis straight E subscript straight R minus straight i straight capital gamma over 2 right parenthesis straight t over fraktur h right square bracket end exponent space for space straight t greater or equal than 0

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