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Probability Amplitude and Interference

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Probability Amplitude and Interference PROBABILITY AMPLITUDE AND INTERFERENCE I. Probability amplitude Suppose that particle A is placed in the infinite square well potential. Let the state of the particle be given by ϕ A and let the system’s energy eigenstates and eigenvalues be given by ψ n and En, respectively, for n = 1, 2, 3, …. A. Write the state of particle A as a sum of the energy eigenstates of the system. Describe how to determine the coefficient of each term in this sum. Ask an instructor to check your expression, and to provide a handout that includes the values of the coefficients in the sum above for a particle A. Write a final expression for the state of particle A in the space below. B. Determine the inner product of the state with itself, ϕϕAA. Show your work. Does your answer agree with what you expect? Explain. C. Suppose you were to measure the energy of particle A. Which value would be the most likely outcome of this measurement? What is the probability of this outcome? Explain. D. Explain why it would be incorrect to say that ψϕnA is the probability that particle A is measured to have energy En. The inner product, ψϕnA, between a state that represents a particle, such as ϕA , and an eigenstate associated with an observable, such as ψ n , is called a probability amplitude. E. Discuss with your group why the term probability amplitude is appropriate for this inner product. 1 F. Suppose that the value of for particle A were changed to . ψϕ2 A − 2 1. Is the probability amplitude associated with n = 2 the same or different? Explain. 2. Is the probability of measuring E2 the same or different? Explain. ü Discuss your answers with an instructor. II. Wave functions ϕA(x) The wave function for particle A, ϕA (),x is shown at right. A. Explain how the wave function is related to the probability density. Describe how to use the probability density to determine the probability that a particle is measured within a small region of width dx. The wave function for particle A can be written as the following inner product: ϕϕAA()xx= , where x is the basis state associated with position x. B. Would it be appropriate to use the term probability amplitude to describe the wave function for particle A? Explain. C. The following statement is incorrect. Identify the flaw(s) in the student’s reasoning. ”When I square the probability amplitude for energy, I get the probability of measuring that energy. Since the wave function is also a probability amplitude, the square of the wave function is also a probability.” D. Write an expression for the state of particle A, ϕA , in terms of the basis states associated with position, x . Explain. ü Discuss your answers with an instructor. III. Interference 11 1 Consider three particles (A, B, and C) described by the states , ϕψψψA =+32124 + 6 11 1 111 , and i . ϕψψψB =32124−+ 6 ϕψC =+326124 ψψ + A. Predict (without sketching) whether the wave functions associated with each of these three states will be the same or different. Briefly explain your reasoning. B. Predict (without sketching) whether the probability densities associated with each of these three states will be the same or different. Briefly explain your reasoning. C. In the space below, write an expression for the probability density for particle B. Show your work. D. Consider the student discussion below. Student 1: ”I know that I have to square the wave function, ϕϕBB()xx= , to get the 2 probability density. This gives me an expression like ϕϕϕBBB()xxx= , which is just equal to ϕϕBB since xx is the identity operator.” Student 2: ”I disagree. I think you can get the probability density by just squaring the wave 11222 1 function for each term in the state, which is 32ψψψ124()xxx++ () 6 () for all three particles.” Student 3: ”That’s right. Since we are calculating the inner product of the state with itself, ϕϕBB, we end up with a bunch of terms that look like ψψ12, which are zero.” All three students are incorrect. Identify the flaws in each student’s reasoning. E. Revisit your predictions from the previous page. Do you still agree with them? Explain. F. Ask an instructor for a handout showing the wave function and the associated probability density for each particle. If your predictions were incorrect, resolve any inconsistencies between the handout and your answers on the previous page. Explain. G. Suppose the sign (or the complex phase) of a single term in a quantum state written in the energy basis is changed. Indicate whether or not each of the following would be different. Explain. 1. Probability amplitudes for energy 2. Probability amplitudes for position 3. Energy probabilities 4. The probability density The results above indicate that in quantum mechanics, probability amplitudes are subject to interference. H. Compare the interference between probability amplitudes in quantum mechanics to other examples of interference that you have seen (e.g., for pulses on a spring or for light waves). .
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