6.13

Exam Expectations

• For each of the nine experiments, expect one or two conceptual physics questions.

• Mini experiments may be included.

• Pre-prepared data may be provided for data analysis.

Required Skills

• Skills learned in physics 4A to 4C lab classes are relevant.

• Know how to do a \pm b squared split.

• Proficiency in propagation of error.

• Familiarity with lab skills outlined in a provided file (to be sent this weekend).

  • Includes standard deviation calculations.

Lab Final Details

• Duration: 2.5 hours.

• Involves analyzing data from different experiments.

• Physical apparatus component: Each student gets 5-10 minutes on setups like ESR (Electron Spin Resonance).

  • Time allocation accounts for cycling through all students.

Differential Equations and Solutions

• Order differential equations have two independent, orthogonal solutions.

• Solutions with k \cdot l = 0 are independent.

Physicality of Solutions

• Solutions that go to infinity may not be physical, even if they satisfy the differential equation.

  • Example: Particle in a finite box, where e^{\alpha x} was rejected for positive x because it blows up.

• Accepted solutions must be normalizable, providing a definite value at specific coordinates.

• Normative functions are generally rejected as non-physical because they are not defined at the origin.

Potentials and Solutions

• Normative functions can be helpful when the particle is not at the origin.

• For a potential where the wave function equals zero at the origin (e.g., particle trapped in a spherical shell):

  • The n = 0 solution's probability density does not blow up.

Effective Potential

• Representation:

  \frac{-\hbar^2}{2m} \frac{d^2}{dr^2} + V(r)

• Can be thought of as a one-dimensional problem with an effective potential.

Free Particle Example

• For a free particle, V = 0, so the effective potential is 1/r^2.

• Analogy: One-dimensional problem with potential equal to some constant over x^2.

Classical Mechanics Connection

• If a particle has non-zero angular momentum, it can't be found at the origin.

• If the particle were at the origin, r \times p = 0, violating conservation of momentum.

• For l \neq 0, the probability density must go to zero as r approaches zero.

• For l = 0, the effective potential is equal to the real potential.

Bessel Functions

• Goal: Show that the J_0 function gives a nonzero probability density at the origin, while other Bessel functions give zero.

Approximations

• Cosine approximation for small angles:

  \cos(\theta) \approx 1 - \frac{\theta^2}{2} + \text{higher order terms}

• Taylor expansion used for improved approximation: 1 - \alpha^2 r^2 / r

Probability Density

• Probability calculation:

  |\psi|^2 r^2 dr

• Objective: Show probability density at origin goes to zero, as in classical mechanics.

Small r Behavior

• For small r (small \theta), sine function term:

\frac{\alpha r}{\alpha r} \approx 1

• Terms:

  • \frac{1}{\alpha r}

  • \frac{-\theta^2}{2} = \frac{-\alpha^2 r^2}{\alpha r}

• The \theta^2 term cannot be ignored.

• Resulting term: \alpha \cdot r

• Radial part of the wave function approaches zero as r goes to zero.

Justification for r²

• Volume between r and r + dr is proportional to r^2.

• Larger values of r have an "unfair advantage" due to volume.

• In Cartesian coordinates, each interval dx occupies same amount of space.

• In spherical coordinates, each interval dr occupies more volume for larger r.

• Expect uniform probability density for small values of r.

J₀ Solution

• Examine e^{ikx} + e^{-ikx} (proportional to cosine) and e^{ikx} - e^{-ikx}.

• These solutions resemble a particle emanating from the origin.

• Analogy to a point source of light, like a candle or the sun, emitting uniformly in all directions.

Free Particle Solution

• Free particle solution in x, y, and z:

  c1 e^{ikx x} e^{iky y} e^{ikz z}

Probability Density of J₀

• Probability density is not uniform, but a cosine squared function.

• Squaring J_0 and multiplying by the r^2 geometric factor yields \cos^2(\alpha r), which is not uniform, but has a uniform average value.

• Average value is possibly uniform.