Review 1.) Sketch the particle-on-a-line wavefunction for n = 1, 2, 3, 4. 2.) On

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Review

1.)    Sketch the particle-on-a-line wavefunction for n = 1, 2, 3, 4.

2.)    On the same sketch, add the square of the wavefunction for the n = 1 state and identify the position(s) where the particle is most likely to be found.

3.)    Repeat for n = 2

4.)    For the n = 3 state, identify the nodes (where the wavefunction changes signs).

5.)    What is the probability of finding a particle at a node?

6.)    What is the general relationship between the number of nodes and the quantum number, n?

Probability

Use the particle-on-a-line model to answer these questions.  

Orthonormality

1.)    Provide one example of two functions that are orthogonal.

2.)    Make a sketch of the n=1 and n=2 wavefunctions for the particle-on-a-line model on the same plot.

3.)    How would the sum of the product of these two wavefunctions for x < a/2 compare to the sum of the product of these two wavefunctions for x > a/2?

4.)    Are these two states orthogonal?

Average Values

1.)    Using a grammatically correct sentence, describe what the symbol <x> means.

2.)    Set up the integral to find the expectation value of x for the particle-on-a-line using normalized particle-on-a-line wavefunction.

3.)    After doing the integral, does your answer agree with what you sketched in the review questions?  Hint:

4.)    Does <x> depend on the quantum number n?

Exercises

1.)    Consider the function (30/a5)1/2(ax-x2).  Is this function normalized?

2.)    Can the function in (1) be an wavefunction for the particle-on-a-line model?  Why or why not?

Sketch the particle-on-a-line wavefunction for n = 1, 2, 3, 4. On the same sketch, add the square of the wavefunction for the n = 1 state and identify the position(s) where the particle is most likely to be found. Repeat for n = 2 For the n = 3 state, identify the nodes (where the wavefunction changes signs). What is the probability of finding a particle at a node? What is the general relationship between the number of nodes and the quantum number, n? Probability Use the particle-on-a-line model to answer these questions. Identify the limits of integration for the particle-on-a-line model. Set up a specific expression that when evaluated would indicate whether or not the particle-in-a-line wavefunction is normalized. Do the normalization. Hint: Using the normalized particle-on-a-line wavefunction, evaluate the probability of finding the particle on the left side of the box. Write an expression for the probability of finding the particle either in the area on the left quarter OR right quarter of the box. Orthonormality Provide one example of two functions that are orthogonal. Make a sketch of the n=1 and n=2 wavefunctions for the particle-on-a-line model on the same plot. How would the sum of the product of these two wavefunctions for x a/2? Are these two states orthogonal? Average Values Using a grammatically correct sentence, describe what the symbol means. Set up the integral to find the expectation value of x for the particle-on-a-line using normalized particle-on-a-line wavefunction. After doing the integral, does your answer agree with what you sketched in the review questions? Hint: Does depend on the quantum number n? Exercises Consider the function (30/a5)1/2(ax-x2). Is this function normalized? Can the function in (1) be an wavefunction for the particle-on-a-line model? Why or why not?

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