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Lecture 15: Waves, Probability and Uncertainty
March 25, 2025
Reading Assignment
- Read: Supplementary Reading Ch 3
Objectives
- (Continuing objective) Describe applications of the concepts of quantum mechanics to everyday “real-life” situations.
- Describe how wavefunction, probability density, and probability are related. Be able to interpret and calculate probability and probability per length both graphically and algebraically. From a sketch of a particle's wavefunction, determine in which regions it is most or least likely to be found.
- Use Heisenberg's uncertainty relation to: (a) relate spreads in position, momentum and velocity; (b) determine the minimum average kinetic energy for a confined particle; and (c) give an explanation for the stability of atoms.
- Test by direct substitution possible solutions of the 1-D Schrödinger equation.
Homework
- Wednesday's Assigned Problems:
Supp CH 3: 4, 5, 6, 7, 9, 10, 11, 13, 14, 16
- Monday's Hand-In Problems from Lecture 15:
Supp CH 3: 2, 3, 8, 15, 17
Note: this is only the first half of the hand-in set.
Lecture Materials
- Click here for the Lecture overheads. Answers: CT1 - 3; CT2A - false, CT2B - true, CT2C - false; CT3 - 1; CT4 - 5
- Fourier Simulation. Download the file and then right click on it and select “open”.
Videos of example problems
To see the problem statement, click on the link below. To play the video example, click on the underlined words "Video Demonstration" near the top of the page with the problem statement.- Example with testing solutions to equations by substitution.
- Example with calculating probabilities, given a wavefunction.
- Example with estimating minimum kinetic energy for a confined particle, using uncertainty.
- Example with using normalization (total probability = 1) for a case with constant probablity density.
- Another example with using normalization (total probability = 1) for a case with varying probablity density.
Pre-Class Entertainment
- Orange Crush - R.E.M.
- You are the Sunshine of my Life - Stevie Wonder
- The Leaving of Liverpool - Gaelic Storm
- Could You Be Loved? - Bob Marley
- Middle of the Road - The Pretenders
Assigned Problems Guide
- Supp 3-4: medium. Need to evaluate $|\psi(x)|^2$ and then integrate theresulting polynomial. For part (b), use $|\psi|^2 = \psi \times \psi^*$, where $\psi^*$ is the complex conjugate, which you obtain by changing the sign of any $i$'s.
- Supp 3-5: medium. Part (a) is straight up Heisenburg uncertainty relation. You can find $\hbar$ in SI units in your inside front cover. For part (b) divide the momentum spread by the mass to get the velocity spread.
- Supp 3-6: quick. Straight up uncertainty principle.
- Supp 3-7: medium-long. Application of the uncertainty principle. Convert kinetic energy to $\sigma_p$ and then find bound on $\sigma_x$, for various situations.
- Supp 3-9: medium-short explain problem. Gets at the heart of the motivation for quantum mechanics.
- Supp 3-10: medium-long. Each part (a)-(c) is the same idea: use an upper bound on $\sigma_x$ to find a lower bound on $\sigma_p$, and then use that to estimate the kinetic energy.
- Supp 3-11: medium-short explain problem. Why is the uncertainty principle a big deal in the microscopic world but not such a big deal at the macroscopic scale.
- Supp 3-13: medium. Testing for solutions to equation. Take the derivative first, then plug it in. Can you make it work for all $x$? If so, what values do the parameters need to have?
- Supp 3-14: long. But very important!! Plugging trial solutions into Schr\ouml;inger's equation to see which can be solutions. We'll come back to this problem in a hand-in and in Thursday's class.
- Supp 3-16: medium-long. Also plugging in trial solutions to Schr\ouml;inger's equation. Don't forget the chain rule!