PROBABILITY AMPLITUDE AND INTERFERENCE

Transcription

1 PROILITY MPLITUDE ND INTERFERENCE I. Probability amplitude Suppose that particle is placed i the ifiite square well potetial. Let the state of the particle be give by ϕ ad let the system s eergy eigestates ad eigevalues be give by ψ ad E, respectively, for =, 2, 3,.. Write the state of particle as a sum of the eergy eigestates of the system. Describe how to determie the coefficiet of each term i this sum. sk a istructor to check your expressio, ad to provide a hadout that icludes the values of the coefficiets i the sum above for a particle. Write a fial expressio for the state of particle i the space below.. Determie the ier product of the state with itself, ϕ ϕ. Show your work. Does your aswer agree with what you expect? Explai.

2 C. Suppose you were to measure the eergy of particle. Which value would be the most likely outcome of this measuremet? What is the probability of this outcome? Explai. D. Explai why it would be icorrect to say that ψ ϕ is the probability that particle is measured to have eergy E. The ier product, ψ ϕ, betwee a state that represets a particle, such as ϕ, ad a eigestate associated with a observable, such as ψ, is called a probability amplitude. E. Discuss with your group why the term probability amplitude is appropriate for this ier product. F. Suppose that the value of ψ2 ϕ for particle were chaged to 2.. Is the probability amplitude associated with = 2 the same or differet? Explai. 2. Is the probability of measurig E 2 the same or differet? Explai. ü Discuss your aswers with a istructor.

3 II. Wave fuctios The wave fuctio for particle, ϕ ( x ), is show at right.. Explai how the wave fuctio is related to the probability desity. ϕ (x) Describe how to use the probability desity to determie the probability that a particle is measured withi a small regio of width dx. The wave fuctio for particle ca be writte as the followig ier product: ϕ ( x) = x ϕ, where x is the basis state associated with positio x.. Would it be appropriate to use the term probability amplitude to describe the wave fuctio for particle? Explai. C. The followig statemet is icorrect. Idetify the flaw(s) i the studet s reasoig. Whe I square the probability amplitude for eergy, I get the probability of measurig that eergy. Sice the wave fuctio is also a probability amplitude, the square of the wave fuctio is also a probability. D. Write a expressio for the state of particle, ϕ, i terms of the basis states associated with positio, x. Explai. ü Discuss your aswers with a istructor.

4 III. Iterferece Cosider three particles (,, ad C) described by the states ϕ = ψ + ψ + ψ ϕ = ψ ψ + ψ, ad ϕ = ψ + i ψ + ψ. C,. Predict (without sketchig) whether the wave fuctios associated with each of these three states will be the same or differet. riefly explai your reasoig.. Predict (without sketchig) whether the probability desities associated with each of these three states will be the same or differet. riefly explai your reasoig. C. I the space below, write a expressio for the probability desity for particle. Show your work. D. Cosider the studet discussio below. Studet : I kow that I have to square the wave fuctio, ϕ ( x) probability desity. This gives me a expressio like is just equal to ϕ ϕ sice x x is the idetity operator. = x ϕ, to get the 2 ( x) = x x, which ϕ ϕ ϕ Studet 2: I disagree. I thik you ca get the probability desity by just squarig the wave fuctio for each term i the state, which is three particles ψ() x ψ 3 2 2() x ψ 6 4() x + + for all Studet 3: That s right. Sice we are calculatig the ier product of the state with itself, ϕ ϕ, we ed up with a buch of terms that look like ψ ψ 2, which are zero. ll three studets are icorrect. Idetify the flaws i each studet s reasoig.

5 E. Revisit your predictios from the previous page. Do you still agree with them? Explai. F. sk a istructor for a hadout showig the wave fuctio ad the associated probability desity for each particle. If your predictios were icorrect, resolve ay icosistecies betwee the hadout ad your aswers o the previous page. Explai. G. Suppose the sig (or the complex phase) of a sigle term i a quatum state writte i the eergy basis is chaged. Idicate whether or ot each of the followig would be differet. Explai.. Probability amplitudes for eergy 2. Probability amplitudes for positio 3. Eergy probabilities 4. The probability desity The results above idicate that i quatum mechaics, probability amplitudes are subject to iterferece. H. Compare the iterferece betwee probability amplitudes i quatum mechaics to other examples of iterferece that you have see (e.g., for pulses o a sprig or for light waves).