Yobinori Quantum Mechanics

• In the PHILUN3601 Metaphysics course, we are dealing with quantum mechanics, but I’m worried because there is no math involved at all.

  • So, I thought I would like to start with something simple for now.

• 2

  • Assumptions:
    • $E={\hslash }^2/2m$
    • Waves can be represented as $\sin 2\pi (x/\lambda -vt)$.
    • $\nu =E/h$, $\lambda =h/p$
      • Relating the energy of an object to frequency and the momentum to wavelength.
      • I remember doing this in IB Physics, but where is the basis for it? (blu3mo)
        • Experimental facts?
        • https://en.wikiversity.org/wiki/De_Broglie_wavelength
        • Ah, it’s a prediction that the same equation as for photons would hold for other particles as well.
          • The dimensions also match.
    • From there, the equation constructed by Schrodinger equation using intuition.
      • I remember doing something similar in PHYS1602 Physics, II: Thermodynamics, Electricity, and Magnetism.
      • There are some logical leaps, but once it is constructed, the equation holds.
    • But still, intuition is amazing. The power of theoretical physics.
  • I can’t really understand the operatorization of physical quantities, but I guess I will understand it eventually.

3

• $\hat{H}$ is an operator for φ.
  • It contains ${\nabla }^2$.
• $\hat{H}$ has multiple pairs of “eigenenergy, eigenfunction” that satisfy it.
  • $\hat{H}{\psi }_n=E_1{\psi }_n$
    • The left side is a partial derivative, and the right side is multiplying by E.
    • It just becomes E times the partial derivative, so there are multiple psi’s that satisfy it.

8

• Consider a linear combination of eigenfunctions.
  • It’s like composing a wave function by combining the smallest units (each eigenfunction).
  • When measuring energy with that wave function, the energy corresponding to one of the eigenfunctions is measured.
  • It’s similar to calculating probabilities in isotopes. (blu3mo)
• For example, if a wave function is created using only one eigenfunction, the measured energy can be narrowed down to one value.