20. In quantum mechanics, the wave function of a particle is determined by the Schrödinger equation. Assume a particle of mass m is placed at a one-dimension infinite potential well that the potential energy U = 0 at the region −L/2 ≤ x ≤ L/2, and U = ∞ when x ＜ −L/2 or x ＞ L/2. Solve the wave function of the particle.

(Note: Ψ₀ is a constant determined by the normalization condition, and n is an integer.)
(A)
(B)
(C)
(D)
(E)

1. A block of mass M hangs on a nail as shown in the figure. A bullet of mass m hits the block horizontally and embeds in it. Find the minimum speed v of the bullet that the block can turn around the nail. (A) ((m+M)/m)√(Lg) (B) ((m+M)/m)√(2Lg) (C) ((m+M)/m)√(3Lg) (D) ((m+M)/m)√(4Lg) (E) ((m+M)/m)√(5Lg)

2. Consider a monochromatic electromagnetic plane wave. The amplitude of its electric field is E₀. What is the cycle-averaged intensity of the wave? (A) (B) (C) (D) (E)

3. As shown in the figure, a conical pendulum consists of an object which moves in a trajectory of a horizontal circle. Find the period T of this motion relative to the angle θ, the string length L, and the gravitational acceleration g. (A) (B) (C) (D) (E)

4. Consider the following processes of an ideal gas system. Which one keeps the entropy of the system constant? (A) Quasi-static adiabatic expansion from volume V to volume 2V. (B) Quasi-static isothermal expansion from volume V to volume 2V. (C) Adiabatic free expansion from volume V to volume 2V. (D) Isothermal free expansion from volume V to volume 2V. (E) Quasi-static isothermal compression from volume V to volume (1/2)V.

5. Consider a thermodynamic system and its surroundings. What is the correct description of a reversible process? (A) A reversible process must be a quasi-static process. (B) During a reversible process, the entropy of the system must keep constant. (C) During a reversible process, the entropy of its surroundings must keep constant. (D) During a reversible process, the temperature of the system must keep constant. (E) During a reversible process, the temperature of its surroundings must keep constant.

6. Cable A and cable B are made up of the same material and of the same length. The cross-section area of cable A is twice that of cable B. When the same electric potential difference is maintained across both cables, which of the following statement is TRUE? (A) Both cables carry the same current density. (B) The current density in cable B is twice as great as the current density in cable A. (C) The current density in cable A is twice as great as the current density in cable B. (D) Both cables carry the same current (E) The current in cable B is twice as great as the current in cable A.

7. A parallel-plate capacitor is connected to a battery at constant potential difference, and allowed to charge up. Put a slab of dielectric material between the plates of the capacitor while the capacitor is still connected to the battery. Which one of the following is TRUE? (A) The energy stored in the capacitor has decreased (B) The energy stored in the capacitor has no changed. (C) The voltage across the capacitor has decreased. (D) The charge on the capacitor has decreased. (E) The charge on the capacitor has increased.

8. A series RC circuit is connected across an ideal DC voltage source through an open switch. The switch is closed at time t = 0 sec. Which of the following statements regarding the circuit is correct? (A) The potential differences across the resistor and the capacitor are always equal. (B) The potential difference across the resistor is always greater than that across the capacitor. (C) The potential difference across the capacitor is always greater than that across the resistor. (D) Once the capacitor is essentially fully charged, the potential difference across the capacitor becomes zero. (E) Once the capacitor is essentially fully charged, the potential difference across the resistor becomes zero.

9. A charged particle is moving in a uniform magnetic field. Its initial velocity is perpendicular to the magnetic field. Which one of the following statements is TRUE? (A) The particle's kinetic energy is changing with time. (B) The radius of the particle's circular path is independent of its speed. (C) The cyclotron frequency of the particle's circular motion is independent of its speed. (D) The cyclotron frequency of the particle's circular motion is independent of its mass. (E) The cyclotron frequency of the particle's circular motion is independent of the magnetic field.

10. Consider an electrical power generator. If you lower the electrical resistance connected across its output terminals while turning the generator at a constant speed, which one of the following statements is TRUE? (A) The peak emf is decreased. (B) The peak output power is decreased. (C) The current flow through the resistor is decreased. (D) The generator gets harder to turn. (E) None of the above.

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