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Chapter 8 The quantum theory of motion Confined motion in an infinite square well in one dimension Schrodinger equation 2 d 2ψ(x) Eψ ( x) 2 2m dx (x) A sin (kx), General solution Boundary conditions 2L , n 1,2,3, n Normalized wave functions n(x) 2 nx sin , L L n 1,2,3, 2 2mE k • A wave function is specified by the quantum number n. • The wave function n(x) has n-1 nodes where the probability of finding the particle is zero. • In a state of wave function with more nodes, the particle has higher kinetic energy. • The probability density to find the particle is nonuniform within the well and is separated by the nodes. Energy levels of an infinite square well • • • • • • • The allowed energy levels are quantized. The allowed lowest energy is not zero and this level is the ground state. Energy level 2k 2 n2h2 E 2m 8mL2 n 1,2,3, The smaller the mass of the particle is, the larger E is. Quantum effect is important for particles of very small mass. The larger the width of the square well, the smaller E is. Quantum effect is important for particles in highly confining regions. E is proportional to h2. This is why quantum effect is not observed by objects in daily life but by electrons in atoms and molecules. Energy difference between two adjacent levels E En 1 En h2 ( 2n 1) 8mL2 Confined motion in an infinite potential well in two dimensions • Position operator xˆ x, yˆ y if 0 x L1 0, and 0 y L 2 V ( x, y ) , otherwise. • Momentum operator pˆ x , pˆ y i x i y • Energy operator Hˆ i t L1 and L2 are the lengths of the box in the x and y directions, respectively. Time-independent Schrodinger’s equation 2 2ψ 2ψ 2 2 Eψ 2m x y The wave function is a function of x and y. Two-dimensional wave functions Separation of variables ψ ( x, y ) X ( x)Y ( y ) (n1,n2)=(1,1) (n1,n2)=(2,1) Solution n x 2 sin 1 , n1 1,2,3, L1 L1 n2 y 2 , n2 1,2,3, Yn2 ( y ) sin L2 L2 X n1 ( x) (n1,n2)=(1,2) n1x n2 y 4 sin , ψ n1 ,n2 ( x, y) sin L1L2 L1 L2 n1 1,2,3, , n2 1,2,3, A wave function is specified by two quantum numbers n1 and n2. (n1,n2)=(2,2) Energy levels and degeneracy • Energy levels kx • For a potential well in a square region, the ground state is (n1,n2)=(1,1) with the lowest energy n1 n , ky 2 L1 L2 En1 , n2 2 k x2 k y2 E1,1 • The states of (n1,n2)=(2,1) and (n1,n2)=(1,2) have the same energy as 2m h 2 n12 n22 2 2 8m L1 L2 n1 1,2,3, E1, 2 E2,1 5h 2 8mL2 • The two states are said to be degeneracy. n2 1,2,3, If L1=L2 =L, the potential well in 2D is in a square region of length L. h2 4mL2 h2 2 2 En1 , n2 n n 1 2 8mL2 n1 1,2,3, , n2 1,2,3, • Degeneracy: States of different wave functions have the same energy. • If N wave functions have the same energy, this energy level is said to be N-fold degeneracy. Confined motion in an infinite potential well in three dimensions • Position operator xˆ x, yˆ y, zˆ z • Momentum operator pˆ x , pˆ y , pˆ z i x i y i z ̂ P i Gradient operator iˆ ˆj kˆ x y z if 0 x L1 , 0, 0 y L , 2 V ( x, y , z ) and 0 z L3 , , otherwise. L1, L2 and L3 are the lengths of the box. Time-independent Schrodinger’s equation 2 2 is a function of ψ Eψ x, y and z. 2m Laplacian operator 2 2 2 2 2 2 2 x y z Three-dimensional wave functions Separation of variables ψ ( x, y, z ) X ( x)Y ( y) Z ( z ) Solution n1x n2 y n3z 8 , sin sin ψ n1 ,n2 ,n3 ( x, y, z ) sin L1L2 L3 L1 L2 L3 n1 1,2,3, , n2 1,2,3, , n3 1,2,3, A wave function is specified by three quantum numbers n1,, n2 and n3. Energy levels and degeneracy • Energy level n32 h 2 n12 n22 En1 , n2 , n3 2 2 2 8m L1 L2 L3 n1 1,2,3, , n2 1,2,3, , E1,1,1 3h 8mL2 • The three states of (n1,n2,n3)=(2,1,1), (1,2,1) and (1,1,2) have the same energy as n3 1,2,3, If L1=L2 =L3=L, the 3D potential is a box. En1 , n2 , n3 • For a potential well in a 3D box, the ground state is (n1,n2,n3)=(1,1,1) with the energy 2 h2 n12 n22 n32 2 8mL E2,1,1 E1, 2,1 E1,1, 2 3h 2 4mL2 • The three states are said to be degeneracy. So, the next energy level is 3-fold degeneracy. Quantum tunneling • • Classical mechanics It is impossible for a particle to surmount over a barrier with potential energy high than its kinetic energy. Quantum mechanics If the barrier is thin and the barrier energy is not infinite, particles have the probability to penetrate into the potential region forbidden by classical mechanics. This is called quantum tunneling. The transmission probably of quantum tunneling generally decays exponentially with the thickness of the barrier and with the square root of the particle mass. V: Potential energy of a barrier E: Kinetic energy of a particle. Left-hand side Right-hand side Transmission probability Transmission probability T is defined as the probability of the particle tunneling through the barrier. The wave function decays exponentially inside the forbidden region but is not zero. | A |2 T | A |2 A and A′ are the amplitudes of the incident and transmitted waves, respectively. Rectangular potential barrier with a thickness L e κL e T 1 16ε (1 ε ) κL 2 1 E / V 1 2m(V E ) 1 , L T 16ε (1 ε )e 2 κL E<V E>V Application of quantum tunneling: Scanning tunneling microscopy Nobel prize in physics 1986 Ca atoms on a GaAs surface Visualization of the reaction SiH3 →SiH +H2 on a Si surface http://www.almaden.ibm.com/vis/stm/stm.html Vibrational motion in a parabolic potential In classical mechanics, a particle attaching to a spring that obeys the Hook’s law vibrates back and forth. The vibrational motion is called harmonic oscillations. The stretching and bending motions of atoms in a molecule is described as harmonic oscillations. Schrodinger’s Eq. 2 d 2ψ(x) 1 2 kx ψ ( x) Eψ ( x) 2 2m dx 2 Boundary conditions ψ ( x) and ψ ( x) 0, as x . Wave function of the ground state 1/ 4 V ( x) 1 kx2 2 d 2x m kx 0 dt 2 x(t ) x0 cos( t ) k m k: Force constant, : Angular frequency 1 ψ 0 ( x) 2 1 x 2 exp 2 Wave function of the first excited state 1/ 4 4 ψ1 ( x) 2 1 x 2 x exp 2 1/ 4 2 mk Wave functions of a harmonic oscillator • The wave functions are specified by the vibrational qunatum number v = 0, 1, 2, 3, …, which is a non-negative integer. • The wave function of the v-th state has v nodes. • The mean displacement <x> in any state is zero, but the mean square displacement <x2> = (v +1/2)2. • The wave functions extend beyond the turning points of the parabolic potential in classical mechanics. So, the particle has a probability to penetrate out of the potential with an exponential decay in distance. • In quantum mechanics, a particle in a double-well potential may tunnel through the forbidden region in classical mechanics. Energy levels of a harmonic oscillator 1 E 2 0 ,1, 2 , 3, E E 1 E • The lowest energy level (v = 0) is E0 = ћ/2, which is the zero-point energy of the oscillator. • The energy levels are uniform spacing of ћ. • The energy levels follow the Planck’s hypothesis. • The higher quantum number a state of the oscillator, the more number of nodes in its wave function and the higher the particle energy is. Rotation in two dimensions Kinetic energy and angular momentum E J2 J2 2 I 2mr 2 J pr mvr I mr 2 The rotational motion of a particle on a plane is described by angular momentum J. I is the moment of inertia of the particle about the rotational axis. Quantization for two-dimensional rotation Quantization of angular momentum (Bohr’s rule) for a fixed radius 2r n, n 0,1,2,3, h nh p J n 2r Wave functions J z ml ml 0, 1, 2,... ψ ml () 1 eiml 2 Energy levels ml2 2 Eml 2I E Eml 1 Eml ( 2ml 1) 2 2I The energy levels are quantized with double degeneracy for the excited states (ml ≠ 0) and nondegeneracy for the ground state (ml = 0). Rotation in three dimensions Spherical polar coordinates In quantum mechanics, the wave function of the particle r, q, is a solution of the Schrodinger’s equation and subject to two boundary conditions. (r , q, 2) (r , q, ) (r , q, ) (r , q, ) In classical mechanics, the rotation of a particle in 3D is on the surface of a sphere, with its position specified by the polar coordinates (q, ), where 0 ≤ q ≤ and 0 ≤ 2. Quantization in angular momentum L2 l (l 1) 2 , l 0,1, 2, 3, l: Orbital angular momentum quantum number Orbital angular momentum and rotational kinetic energy L mr v L2 E 2mr 2 Energy levels l (l 1) 2 El 2mr 2 Angular momentum in three dimensions • The angular momentum about the z-axis is also quantized. Representation of wave functions for ml=0 Lz ml ml 0, 1, .... ,l • Each level is specified by two quantum number l and ml. • l : Orbital angular momentum quantum number • ml : Magnetic quantum number • Both quantum numbers are values of integer. • l has non-negative integral values, 0, 1, 2,…. • ml is limited to 2l+1 values from –l, -l+1, …, l-1, 1. Vector model for angular momentum l =2 The energy levels El,ml of each orbiatl quantum number l are (2l+1)-fold degeneracy. Exercises • 8A.5(a), 8A.12(b), 8A.13(b) • 8B.3(a), 8B.4(a), 8B.6(a), 8B.9(a) • 8C.4(a), 8C.5(a), 8C.6(a)