Particle in 3D box
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1) The document discusses solving the Schrodinger equation for an electron confined to a three dimensional box. 2) It shows that using separation of variables, the Schrodinger equation can be broken into three one dimensional equations that are each equivalent to the particle in a one dimensional box. 3) The solutions for the wavefunctions and energy levels are presented, with the energy levels increasing with higher quantum numbers and becoming degenerate with multiple combinations of the quantum numbers.
Particle in 3D box
- 1. Dr. Pradeep Samantaroy Department of Chemistry Rayagada Autonomous College, Rayagada pksroy82@gmail.com; pksroy82@yahoo.in 9444078968
- 2. V (x,y,z) = 0 V (x,y,z) = ∞ O < x ≤ Lx O < y ≤ Ly O < z ≤ Lz Understanding the Case Quantum Approach Prepared by Dr. Pradeep Samantaroy
- 3. Let’s solve the case…! )()()( )()()( 2 2 2 2 2 2 22 rErrV dz rd dy rd dx rd m The Schrodinger’s equation for the electron in three dimensional box can be represented as )()()()( 2 2 2 rErrVr m It can also be represented as Prepared by Dr. Pradeep Samantaroy
- 4. Outside and on the Edges of Box Applying boundary conditions outside the box Er m *)( 2 2 2 0 2 Hence, the probability of finding electron outside the 3D box is zero. Boundary conditions ψ(0,y,z) = ψ(Lx,y,z) = 0 Ψ(x,0,z) = Ψ(x,Ly,z) = 0 ψ(x,y,0) = ψ(x,y,Lz) = 0 Prepared by Dr. Pradeep Samantaroy
- 5. Inside the box )( )()()( 2 2 2 2 2 2 22 rE dz rd dy rd dx rd m 0)( rVSince To solve this equation, separation of variable can be applied. ψ(x,y,z)= X(x) Y(y) Z(z) Now we can see that each function has its own variable: X(x) is a function for variable x only Y(y) is a function of variable y only Z(z) is a function of variable z only Prepared by Dr. Pradeep Samantaroy
- 6. Now substituting the value of Ψ (x,y,z) in the Schrodinger’s equation and then dividing the whole by xyz, E dz zZd zZdy yYd yYdx xXd xXm 2 2 2 2 2 22 )( )( 1)( )( 1)( )( 1 2 Now the final equation can be represented as Prepared by Dr. Pradeep Samantaroy
- 7. We also partition the total energy into its rightful components: E = Ex + Ey + Ez Now each component can be separated as Does this equation looks similar to what we have studied already!!! It is the case of particle in 1D box. Hence the solution of x component is )( 2 )(2 2 2 xXE dx xXd m x x x x x L xn L sin 2 Prepared by Dr. Pradeep Samantaroy
- 8. )( 2 )(2 2 2 yYE dy yYd m y y y y y L yn L sin 2 Similarly the y and z component can be represented as )( 2 )(2 2 2 zZE dz zZd m z z z z z L zn L sin 2 Prepared by Dr. Pradeep Samantaroy
- 9. Therefore the final solution for a particle in 3D box can be represented as z z zy y yx x x L zn LL yn LL xn L r sin 2 sin 2 sin 2 )( z z y y x x zyx L zn L yn L xn LLL r sinsinsin 8 )( z z y y x x L zn L yn L xn V r sinsinsin 8 )( or or Prepared by Dr. Pradeep Samantaroy
- 10. Energy 2 22 8 x x x mL hn E Now the energy of the x component can be written as Similarly the energy of the y and z component can be written. Now the total energy can be written as 2 22 2 22 2 22 888 z z y y x x zyx mL hn mL hn mL hn EEEE 2 2 2 2 2 22 8 z z y y x x L n L n L n m h E Prepared by Dr. Pradeep Samantaroy
- 11. If the 3D box is a cube? If the 3D box is a cube then Lx = Ly = Lz = L L zn L yn L xn V r zyx sinsinsin 8 )( 2 2 2 2 2 22 8 L n L n L n m h E zyx 222 2 2 8 zyx nnn mL h E Prepared by Dr. Pradeep Samantaroy
- 12. Degeneracy in a 3D cubical box Can any one or all of nx, ny, nz values be zero?? Ground state energy can be calculated using nx= 1, ny= 1, nz= 1 2 2 222 2 2 111 8 3 111 8 mL h mL h E Zero Point Energy Prepared by Dr. Pradeep Samantaroy
- 13. Degeneracy in a 3D cubical box For the first excitation state energy can be calculated using following combination of quantum numbers (nx,ny, nz) = (2,1,1) or (1,2,1) or (1,1,2) 2 2 112121211 8 6 mL h EEE Corresponding to these combinations of (nx, ny, nz), three different wave functions and three different states are possible. Hence, the first excited state is said to be three-fold or triply degenerate. Prepared by Dr. Pradeep Samantaroy
- 14. nx , ny , nz nx 2 + ny 2 +nz 2 Energy Degeneracy (111) 3 3h2/8mL2 1 (211), (121), (112) 6 6h2/8mL2 3 (221), (212), (122) 9 9h2/8mL2 3 (311), (131), (113) 11 11h2/8mL2 3 222 12 12h2/8mL2 1 (123), (132), (213), (231), (312), (321) 14 14h2/8mL2 6 Degeneracy in a 3D cubical box-Summary Prepared by Dr. Pradeep Samantaroy
- 15. 12 9 6 3 14 11 (1,1,1) (1,2,1)(2,1,1) (1,1,2) (2,1,2)(2,2,1) (1,2,2) (1,3,1)(3,1,1) (1,1,3) (2,2,2) (1,3,2)(1,2,3) (2,1,3) (3,1,2)(2,3,1) (3,2,1) Energyinunitsofh2/8mL2 Prepared by Dr. Pradeep Samantaroy