Understanding Quantum Mechanics: Wave Functions and Particle Behavior
Explore the properties of wavefunctions, particle behavior in a box, Schrödinger's equation, boundary conditions, and normalized wave functions in one-dimensional quantum mechanics. Learn about the key principles and mathematical concepts that govern the behavior of particles in confined spaces.
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Presentation Transcript
Unit IV, Part 4 Quantum Mechanics BSc III
Properties of the Wavefunction and its First Derivative must be finite for all x must be single-valuedfor all x must be continuous for all x
Particle in a box Since the walls are impenetrable, there is zero probability of finding the particle outside the box. Zero probability means: (x) = 0, for x 0 and x L The wave function must also be 0 at the walls (x = 0 and x = L), since the wavefunction must be continuous Mathematically, (0) = 0 and (L) = 0
The motion of the electron in one dimensional box can be described by the Schr dinger's equation. 2 2 d dx mE V + ] = [ 0 d (1) 2 2 Inside the box the potential V =0 2 dx 2 d m + ] = ) 2 ( [ 0 E 2 2
2 d dx + 2 = 0 (3) k 2 2 mE where k = 2 ., 2 The solution to equation (3) can be written as = + ) 4 ( ( ) sin cos x A kx B kx Where A,B and k are unknown constants and to calculate them, it is necessary To apply boundary conditions.
When x = 0 then = 0 i.e. ||2 = 0 ------(i) x = L = 0 i.e. | |2 = 0 ------(ii) Applying boundary condition (i) to equation (4) A sin k(0) + B cos k(0) = 0 B = 0 Substitute B value equation (4) (x) = A sin kx
Applying second boundary condition to equation (4) = sin + 0 sin ) 0 ( cos A kL = 0 kL 0 A sin kL = kL = kL n n = k L Substitute k value in equation (5) x ( ( ) n x = ) sin A L To calculate unknown constant A, consider normalization condition.
L 2 = Normalization condition ( ) 1 x dx 0 L n x = 2 2 sin [ dx ] 1 A L o L 1 2 n x = 2 1 [ 2 cos[ dx ] 1 A L o 2 2 A 2 L nx = L 0 [ sin ] 1 x 2 n L 2 A 2 = 1 L = 2 / A L
The normalized wave function is n x = 2 / sin L n L Dimensiona Three case l For n z n x n y = 3 3 L 2 ( / ) sin sin sin 1 L 2 L L n = + + 2 2 1 2 2 2 3 n n n n The wave functions n and the corresponding energies En which are often called Eigen functions and Eigen values, describe the quantum state of the particle.
Energy Calculation 2 mE = 2 since , k 2 2 2 k = E 2 wherem ., n h = = & k 2 h L 2 2 n 2 L = E 2 m 2 2 n h = E 2 8 mL
The electron wave functions nand the corresponding probability density functions |n|2 for the ground and first two excited states of an electron in a box are shown in Figs. (a) and (b) respectively.
References 1. Arthur Beiser, Concepts of Modern Physics, New York: McGraw-Hill, 2002. 2. Nouredine Zettili, Quantum Mechanics: Concepts and Applications, John Wiley and Sons, 2001. 3. Satya Prakash, Quantum Mechanics, Pragati Prakashan, 2018.
