### Shortcut Methods

**Modern Physics problem-solving shortcuts and tricks**

**1. Photoelectric effect**

- To calculate the threshold wavelength, use: (\lambda_{th}=\frac{hc}{\phi})
- For maximum kinetic energy of photoelectrons: (K_{max}=h\nu-\phi)

**2. Compton effect**

- To calculate the change in wavelength, use: (\lambda’ - \lambda = \frac{h}{mc}(1-\cos \theta)))
- Scattered X-ray energy: (E’=\frac{hc}{\lambda’})

**3. Pair production**

- Minimum energy required: (E_{min}=2mc^2)
- Total energy of electron-positron pair: (E_{total}=2mc^2+K_e+K_p)

**4. Radioactive decay**

- Half-life calculation: (t_{1/2}=\frac{ln2}{\lambda})
- Activity: (A=-\frac{dN}{dt})
- Amount remaining: (N(t)=N_0e^{-\lambda t})

**5. Nuclear reactions**

- Released energy: (E=mc^2(1-\frac{m_{final}}{m_{initial}}))
- Balancing equations: Conserve nucleons and charge

**6. Quantum mechanics**

- Solve the Schrödinger equation: Use appropriate techniques for different potentials
- Calculate wave function: (\psi(x,t))
- Energy levels: Solve for allowed (E) in the Schrödinger equation

**7. Quantum entanglement**

- Understand the concept and applications in cryptography and computing