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Lecture 4. : The Free Particle The material in this lecture covers the following in Atkins. 11.5 The informtion of a wavefunction (a) The probability density Lecture on-line Free Particle (PDF) Free Particle (HTML) Free Particle (PowerPoint) QuickTime™ and a GIF decompressor are needed to see this picture. Tutorials on-line The postulates of quantum mechanics (This is the writeup for Dry-lab-II)( This lecture does not cover any specific postulate) Standing Wave (animation) (a must) The wave Packet as superposition of plane waves (annimation) (a must) A complete walk-through the free paricle (a must) The Development of Classical Mechanics Experimental Background for Quantum mecahnics Early Development of Quantum mechanics Audio-visuals on-line The dual nature of matter (Quick Time movie 9 MB from Wilson group, *** ) Linear polarized light ( a wave function in 1-D would propagate in a similar way) (1 MB Quick time movie from the Wilson Group, *****) Circular polarized light ( ( a wave function could propagate in a similar way) (6 MB Quick time movie from the Wilson Group, *****) Slides from the text book (From the CD included in Atkins ,**) The Classical Hamiltonian Review Consider a particle of mass m that is moving in one dimension. Let its position be given by x X QuickTime™ and a GIF decompressor are needed to see this picture. O In classical mechanics the state of a particle moving in 1 - D with the potential energy V(x,t) is determined completely from the initial conditions at t= t o : dV d 2x x(t o ) ; v(t o ) and m 2 (Newtons Law) dx dt The Classical Hamiltonian Review The expression for the total energy in terms of the potential energy and the kinetic energy given in terms of the linear momentum: E Ekin Epot p2 V(r ) 2m is called the Hamiltonian: p2 H V(r ) 2m Quantum Mechanics Review X QuickTime™ and a GIF decompressor are needed to see this picture. O The state of the same system in quantum mechanics is determined by the time dependent Schrödinger equation: (x,t) ˆ H(x,t) i t according to postulate 6 (x, t) (x, t) V(x, t)(x, t) 2 i t 2m x 2 2 The wavefunction (X,t) contains all the information about the system according to postulate 1 Stationary States Review The total wavefunction for a one-dimentional particle in a potential V(x) is given by E (x, t) f(t)(x) AExp[i t](x) Where (x) is determined from the time independent Schrödinger equation ˆ (x) E(x) H Or : 2 2 (x) (x)V(x) E(x) 2 2m x Energy of system Quantum Mechanical Principles..the Free Particle The equation for (x) is given by 2 (x) 2 (x)V(x, t) E (x) 2m x Let us now assume that V(x) = 0 2 In that case : (x) 2 E (x) 2m x 2 2 Quantum Mechanical Principles..the Free Particle (x) 2 E (x) 2m x 2 2 with the general solution : (x) Aexp ikx Bexp ikx since : 2 [ Aexpikx B expikx ] k 2 2 ikx ikx [Aexp Bexp ] 2 2m 2m x 2 (x) or : 2 E (x) 2m x 2 2 2 2 k E 2m Quantum Mechanical Principles..the Free Particle 2 2 (x) 2 2 k E (x) E 2 2m x 2m (x) Aexp ikx Bexp ikx Since the particle only has kinetic energy we must have E 2 2 k 2m 2 p 2m or : p = k Quantum Mechanical Principles..the Free Particle i ikx ikx (x, t) Exp Et Aexp Bexp ) Time face factor f (t). Note f (t) f (t) 1 * Has both real part Acos kx Bcos kx and imaginary part i( Asinkx Bsinkx) Quantum Mechanical Principles..the Free Particle We get for the probability density: i i (x, t)(x, t) Exp Et Exp Et ikx ikx ikx ikx 1 Aexp B exp ) Aexp Bexp ) * Or (x, t)(x, t) A exp * 2 ikx AB(exp i2 kx B exp exp 2 ikx exp ikx i 2kx ) exp ikx Quantum Mechanical Principles..the Free Particle (x, t)(x, t) A exp * 2 ikx AB(exp i2 kx B exp 2 1 exp ikx exp ikx 1 i 2kx exp ikx ) | (x, t) | A B AB[cos2kx i sin 2kx] AB[cos2kx i sin 2kx] 2 2 2 (x, t) (x, t) A B 2 ABcos 2kx * 2 2 Quantum Mechanical Principles..the Free Particle (x, t) (x, t) A B 2 ABcos 2kx * 2 2 B0 i (x, t) AExp Et Expikx i (x, t) AExp Et (cos kx i sin kx) 2 * (x, t) (x, t) A Same probability everywhere Quantum Mechanical Principles..the Free Particle i Et (x, t) AExp Et Expikx AExp i(kx ) (x, t) Acos(kx Et ) iAsin(kx Et ) Re (x, t) Acos(kx QuickTime™ and a GIF decompressor are needed to see this picture. Nodes move to the right Et ) Quantum Mechanical Principles..the Free Particle Let us assume that at : E Thus Acos(kx - Et ) 0 t = to we have : kx - to = 2 At the later time : E E E E t = to t kx - t = kx - t0 t t 2 QuickTime™ and a GIF decompressor are needed to see this picture. E However at : x + t k we have at : to + t that E E E k(x + t) (to t) kx - to = k 2 E The node has traveled x = t k in t Re (x, t) Acos(kx Et ) Quantum Mechanical Principles..the Free Particle E The node has traveled x = t k in t E velocity of the node is x/dt = k k2 2 E p mv v x/dt = k 2m k 2m 2m 2 2 2 k E 2m p = k Re (x, t) Acos(kx QuickTime™ and a GIF decompressor are needed to see this picture. Nodes move to the right Et ) Quantum Mechanical Principles..the Free Particle i ikx ikx (x, t) Exp Et Aexp Bexp ) (x, t) (x, t) A B 2 ABcos 2kx * 2 2 A 0 i (x, t) BExp Et Expikx (x, t) * (x,t) B2 i (x, t) BExp Et (cos(kx) i sin(kx)) Quantum Mechanical Principles..the Free Particle i Et (x, t) AExp Et Expikx AExp i(kx ) (x, t) Bcos(kx Et ) iBsin(kx Et ) Re (x, t) Bcos(kx QuickTime™ and a GIF decompressor are needed to see this picture. Bcos(kx Et ) Nodes move to the left v x/dt = 2 Et ) Quantum Mechanical Principles..the Free Particle AB (x, t) (x, t) A B 2 ABcos 2kx * 2 2 i ikx ikx (x, t) Exp Et Aexp Bexp ) i (x, t) AExp Et (Expikx Expikx ) Quantum Mechanical Principles..the Free Particle i o (x,t) AExp Et (cos kx i sin kx cos kx i sin kx) i o (x,t) 2AExp Et cos kx | o (x, t) |2 2 A2 (1 cos 2kx) | o (x, t) |2 4A 2 cos2 kx) Quantum Mechanical Principles..the Free Particle i o (x,t) AExp Et (Expikx Expikx ) Et Et o (x,t) A(Exp i(kx Exp i(kx ) Et Et o (x,t) A(cos(kx ) i sin(kx ) Et Et A (cos(kx ) i sin(kx ) o (x,t) 2A cos(kx)cos( Et ) i cos(kx)sin( Et ) Quantum Mechanical Principles..the Free Particle QuickTime™ and a GIF decompressor are needed to see this picture. Re (x, t) i Re AExp Et Expikx Et Acos(kx ) Nodes move to the right QuickTime™ and a GIF decompressor are needed to see this picture. Re (x, t) i Re AExp Et Expikx Et Acos(kx ) Nodes move to the left Re (x, t) Acos(kx Et ) Re (x,t) Acos (kx Et Re o (x, t) 2 Acos(kx)cos( ) Re Re QuickTime™ and a GIF decompressor are needed to see this picture. Nodes do not move Et ) What you should learn from this lecture 1. For a free particle [ V(x) = 0] the time- independent Schrödinger equation is given by : 2 2 (x) E(x) 2 2m x with the general solution : (x) A expikx Bexpikx k2 2 and energy E given by E = 2m You should also understand why physical arguments require that the linear momentum must be p = k What you should learn from this lecture 2. For B = 0 ; (x) A expikx This wave function represents a particle moving in the positive x - direction with a constant probability density (x)* (x) = A. We shal later show that the particle has the momentum p= k 3. For A = 0 ; (x) Bexpikx This wave function represents a particle moving in the negative x - direction with a constant probability density (x)* (x) = B. We shal later show that the particle has the momentum p= -k 4. For A = B ; (x) A[expikx expikx ] This wave function represents a particle with a probability density (x) * (x) = 4A2 cos2kx. We shall later discuss what state the particle is in 5. You should understand Figure 11.21 (page 300 in Atkins)