Quick run through on classical mechancis and quantum mechanics
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The slides are designed for my guided study in MSc CUHK. It is about the brief description on classical mechanics and quantum mechanics . Some Slides I got from the slideshare clipboards for better illustration of the ideas in Physics. Thanks to slideshare, I make a milestone on presenting one of the prominent fields in modern physics.
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Quick run through on classical mechancis and quantum mechanics
- 1. Guided Study: A Quick Run Through Classical and Quantum Mechanics Msc Student: Alan Leung Shek Lun Advisor: Professor Hui Pak Ming
- 2. Outline Classical Mechanics (mass-spring system) 1. Newton’s laws of motion 2. Momentum 3. Work-energy Theorem 4. Potential Energy and force (relation) 5. Harmonic Oscillator 6. Lagrangian Mechanics and Hamiltonian Mechanics Quantum Mechanics 1. Schrodinger Equation 2. Typical 1D problems 3. Typical 2D and 3D problems and their key features 4. Quantum Harmonic Oscillator
- 4. Newton’s law of motion At time t=0s, the box is in equilibrium, i.e. Fnet = 0N Displace a distance xm for oscillation The force always acts toward the equilibrium is restoring force. By Hook’s law, F = -kx F µ x Negative sign:pointing in opposite direction
- 5. Newton's law of motion The motion of the box is periodic, like a sinusoidal function, we can be express x(t) = Asin(wt +f), Diiferentiate both sides with respect to t, v(t) = x'(t) = wAcos(wt +f), vmax = wA a(t) = x''(t) = -w2 Asin(wt +f), amax = w2 A A is the boundary condition, x £ A, since it is the internal force that drives the motion.
- 6. Newton’s law of motion m(-Aw 2 sin(wt +f))+ kAsin(wt +f) = 0 k m A = w2 A, w = k m = 2p f Also, it is a form of simple harmonic motion: d2 x dt2 = - k m x = -w2 x By Newton's law of motion F = ma -kx = m d2 x dt2 m d2 x dt2 + kx = 0,which is a second order differential equation The solution to this differential equation is of the form: x(t) = Asin(wt +f)
- 7. Relation of Force and Potential Energy For some conservative forces, Slope = F = - dV(x) dx = -kx At equilibrium point, F = dV(x) dx = 0N When dV(x) dx > 0, F < 0, vice versa. Integrating both sides with respect to x and get, Area = Fò dx = -dV(x)ò = -kxdxò V(x) = - -dV(x) = - -kxdxòò = kx2 2 V(x) µ k The narrowness of the curve depends k.
- 8. Work and Energy Theorem F = d dt (mv) F dx = 1 2 ò mvdx 1 2 ò = m dv dx dt1 2 ò = 1 2 m(v2 2 - v1 2 ) = T2 - T1 = KE F = - dV(x) dx F dx 1 2 ò = - dV(x) 1 2 ò = -V2 +V1 = PE
- 9. Total Mechanical Energy PE = KE -V2 +V1 = T2 -T1 T1 +V1 = T2 +V2 Total mechanical energy = Kinetic Energy + Potential Energy
- 10. Conservation of Energy E = T +V = 1 2 mv2 + 1 2 kx2 = 1 2 m(Aw cos(wt +f))2 + 1 2 k(Asin(wt +f))2 = 1 2 kA2 (cos(wt +f)2 + sin(wt +f)2 ) (∵ w2 = k m ) = 1 2 kA2 , which is a constant. Energy is conserved.
- 11. Boundary Condition When the box is at its maximum point, PE = max, KE = 0J E = PE = 1 2 kA2 Total energy is constant, i.e. harmonic oscillator is a conservative system. When the box is at its equlibrium point, PE = 0J, KE = max E = KE = 1 2 mv2 max
- 12. Phase Diagram of momentum and position E = T +V E = 1 2 mv2 + 1 2 kx2 1 2 kA2 = px 2 2m + 1 2 kx2 (∵ p = mv) 1= px 2 mkA2 + x2 A2 This is an equation of ellipse, px 2 b2 + x2 a2 = 1, a = A2 & b = mkA2 When p = 0, 1 2 kA2 = 1 2 kx2 Þ x = A When x = 0, 1 2 kA2 = px 2 2m Þ px = mkA2
- 17. Conservative law
- 18. Conservative law If the transformation correspond to a symmetry transformation, Lagrangian is invariant, d L = 0 d dt pidqi( ) i å = 0 pidqi( ) i å = pi fi (qi )( ) i å = Q which is a certain quantity, so momentum is conserved.
- 19. In Lagrangian, we have talked about least action of motion, momentum where is ENERGY?
- 20. Change in coordinate dq ®dq +d If d L = 0, L is invariant, the sysmmetric There is conservative law.
- 26. 5 postulates in Quantum Mechanics
- 34. For harmonic oscillator, it can be expressed as
- 36. Quantum Mechanics Typical 1D problem
- 37. A free particle
- 39. Particle in an infinite well
- 40. Finite Potential Well (bound states)
- 43. Simulation The one dimensi onal particle in a Box
- 44. Simulation Superposition States in an infinite square well
- 45. Quantum Mechanics 2D and 3D problems and their key features
- 46. Simulation Energy Eigenfunction of the two dimensional infinite well
- 50. Quantum Harmonic Oscillator Energy level
- 51. When the energy is getting larger, It is more close to Classical phonomenen
Hinweis der Redaktion
- w
- On a smooth surface, a box of mass m is attached to a spring horizontally At time t=0s, the box is in equilibrium There is no external force acting on it To make it oscillate, it can be displaced from rest to the right by xm or to the left by xm. K force constant X displacement
- W is angular frequency Phi is phase angle
- By Newton’s 2nd law of motion, the acceleration is in proportion to the external force acting on it And inversely proportional to the body mass Put first and second derivative into differential equation F is the frequency of the oscillation
- Conservative forces are those for which the work required to move an object from A to B is independent of the path taken. Eg Gravity Electricity Magnetism Spring force Eg of non conservative force Friction Air resistance The conservative force equals the negative(partial) derivative of the potential energy with respect to x. The negative sign means the final potential energy of the system is lower than the initial as the conservative force does work.
- hamiltonian equation
- V(x)