### Shortcut Methods

**Shortcuts and Tricks for Solving Numerical Problems**

**1. Particle Motion:**

- De Broglie Wavelength (λ): $$λ = \frac{h}{mv}$$ where h is Planck’s constant (6.626 x 10^-34 Js), m is the particle’s mass, and v is its velocity.

**2. Photoelectric Effect:**

- Maximum Kinetic Energy (KE_max): $$ KE_{max} = hf - \varphi $$ where h is Planck’s constant, f is the frequency of light, and φ is the work function of the metal surface.

**3. Quantum Mechanics:**

- Schrödinger Equation: $$-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$$ where Ψ is the wave function, ħ is the reduced Planck constant, m is the particle’s mass, V is the potential energy function, E is the total energy, and x is the position.

**4. Nuclear Reactions:**

- Energy Released (Q): $$Q = (\Delta m) c^2$$ where Δm is the mass defect (difference between the masses of reactants and products) and c is the speed of light.

**5. Radioactive Decay:**

- Half-Life (t1/2): $$t_{1/2} = \frac{ln(2)}{λ} $$ where λ is the decay constant.

**6. X-Ray Scattering:**

- Wavelength (λ): $$λ = 2d \sin\theta$$ where d is the lattice spacing and θ is the scattering angle.

**7. Diffraction:**

- Angular Separation (θ): $$θ = \sin^{-1} \left(\frac{mλ}{d}\right)$$ where m is the order of the bright spot, λ is the wavelength of light, and d is the grating spacing.

**8. Relativistic Effects:**

- Relativistic Mass (m_rel): $$m_{rel} = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$ where m_0 is the rest mass, v is the particle’s velocity, and c is the speed of light.

**9. Atomic Energy Levels:**

- Energy Difference (ΔE): $$ΔE = hf $$ where h is Planck’s constant and f is the frequency of the light emitted or absorbed.

**10. Quantum Harmonic Oscillator:**

- Ground-State Energy (E_0): $$E_0 = hf_0 = \frac{1}{2}h\sqrt{\frac{k}{m}}$$ where f_0 is the ground-state frequency, h is Planck’s constant, k is the spring constant, and m is the effective mass.

Remember, these are just a few shortcuts and tricks, and in some cases, detailed calculations might be necessary.