Quantum Motions

Quantum Motions - 3 TYPES OF MOTIONS: TRANSLATION,...

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Quantum Motions 1 3 TYPES OF MOTIONS: TRANSLATION, VIBRATION & ROTATION TRANSLATIONAL MOTION Particle with mass m, free to move in a straight line, between two walls. The walls are impenetrable and infinitely long. The electron in a box is a model of moving molecules and can be used to obtain rough estimates of translational energies. The electron in a box is a model of free electrons in a conjugated molecule and can be used to obtain rough estimates of electronic transitions. V= V=0 V= 0 L x Particle in a one-dimensional box: A particle of mass m, subjected to a potential that is: V=0 within x=0 and x=L V= at x=0 and x=L The particle is confined inside the box.
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Quantum Motions 2 (x) = a 0 cos (kx) + a 1 sin (kx) a general solution. The wavefunction is zero outside the box: (x) = 0 for x 0 and x L thus: (0) = a 0 cos (0) + a 1 sin (0) then a 0 = 0 from the first boundary condition We are left with: (x) = a 1 sin (kx) (L) = a 1 sin (kL) = 0 second boundary condition If a 1 = 0 we don't have a wave. So sin (kL) must be 0 True for kL = , 2 , 3 , . .., n True for k = ( /L), (2 /L), (3 /L), . .., (n /L) We do not include k = 0 because in that case the wave would be 0 for any x. We need a wave that has an n: n (x) = a 1 sin (n x/L) with n=1, 2, . .. (no n=0 !!)
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Quantum Motions 3 Normalize: (from x=0 to L) *(x) (x) dx = 1 a 1 2 (from x=0 to L) sin 2 (n x/ L) dx = 1 change of variable: n x/L = y differential: dy/dx = n /L so dx=L/(n ) dy limits: when x=0, y=0; when x=L, y=n a 1 2 (from y=0 to n ) sin 2 y [ L/ (n )] dy = 1 a 1 2 [ L/(n )] (from y=0 to n ) sin 2 y dy = 1 a 1 2 [ L/(n )] (from y=0 to n ) 1 / 2 [1 - cos (2y) ] dy = 1 since sin 2 (y) = 1 / 2 [1 - cos (2y) ] a 1 2 [ L/(n )] 1 / 2 { (from y=0 to n )dy - (from y=0 to n ) cos (2y) dy } = 1 using: cos ax dx = (1/a) sin ax a 1 2 [ L/(n )] 1 / 2 { n - [ 1 / 2 sin (2n ) - 1 / 2 sin (0) ] = 1 a 1 2 [ L/(n )] 1 / 2 { n - 0 + 0 } = 1 a 1 2 [ L/2] 1 so a 1 = ( 2 / L ) Note that the normalization constant does not depend on n. The maximum amplitude is always the same.
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Quantum Motions 4 1. Wavefunctions for L=1, and x between 0 and 1: ,..., 2 , 1 n L x 0 L x n sin L 2 ) x ( 1-dim particle in a box -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 x Psi(x) n=1 n=2 n=3 n=10 The wavefunction becomes more classical with larger n's: correspondence principle . Fully packed would look like a line (a constant).
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Quantum Motions 5 2 3 2 2 2 2 mL 2 n h L n m 2 E Since is normalized: 2. Energies 1-dim t-independent Schrödinger equation: Two ways of calculating energy. First, using expectation values . Inside the box, V(x)=0: ) x ( E ) x ( ) x ( V ) x ( x m 2 2 2 2 dx ) x ( d ) x ( * m 2 E L 0 2 2 2 L x n cos L n L 2 ) x ( d ) ( ) 1 ( sin ) 1 ( 2 ) ( 2 2 2 2 x L n L x n L n L x d dx ) x ( ) x ( * L n ) 1 ( m 2 E L 0 2 2 L x n sin L 2 ) x (
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Quantum Motions 6 The energy is quantized (depends on n 2 ) Steps have larger increments with increasing value of n. For L=1Å and an electron: 2 3 2 2 mL 2 n h E Second way, solving the Schrodinger equation directly . It can be done because (d 2 /dx 2 ) gives a constant: divide by (x) All this energy is kinetic.
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Quantum Motions - 3 TYPES OF MOTIONS: TRANSLATION,...

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