--%>
Assignment Help - homework help

Schrodinger equation with particle in a box problem.

Three dimensional applications of the Schrodinger equation are introduced by the particle-in-a-box problem.

So far only a one-dimensional problem has been solved by application of the Schrodinger equation. Now the allowed energies and the probability functions for a particle that is free to move in three dimensions are deduced. A molecule of a gas enclosed in a cubic container provides a specific example that is dealt with in the section after the general procedure has been developed.

For any three-dimensional problem, the potential energy is, general, a function of three coordinates. For a cubic potential box, the Cartesian coordinates are convenient. The differential equation that must be solved is now the Schrodinger equation in three dimensions.

1310_Particle in a box.png 

For a "cubic box," the potential function can be expressed in terms of separate x, y, and z components,
98_Particle in a box1.png 

Each of the potential function components for a "particle-in-a-box" is like the one-dimensional potential for a "particle-on-a-line".

For three-dimensional systems, the solution function ψ depends on the three coordinates necessary to locate a point in space. It is often profitable to try to separate such systems into parts, with each part involving only one coordinate. On the basis we try the substitution

ψ (x, y, z) = Ø(x)Ø(y)Ø(z)

Substitution of (2) from (1) gives

1350_Particle in a box2.png 

Division by Ø(x)Ø(y)Ø(z) gives

1180_Particle in a box3.png 

For the equation to be satisfied for all values of x, y and z, each term on the left must equal a component of ε, and we can write

ε = εx + εy + εz

The Schrodinger equation can then be broken down into three identical equations of the type

1394_Particle in a box4.png 

Or

578_Particle in a box6.png 

These equations are identical to that written for one-dimensional problem. The solution to the three-dimensional cubic-box problem is therefore

ψ =  Ø(x)Ø(y)Ø(z)

With

1809_Particle in a box7.png

   Related Questions in Chemistry

  • Q : Nuclear Magnetic Resonance The nuclear

    The nuclear states produced by a magnetic field are studied in nuclear magnetic resonance spectroscopy.The frequency of the radiation that corresponds to the nuclear magnetic energy level spacings and the weakness of the radiation absorption that must be e

    		•
  • Q : Calculating total vapour pressure

    Select the right answer of the question. The vapour pressure of two liquids P and Q are 80 and 600 torr, respectively. The total vapour pressure of solution obtained by mixing 3 mole of P and 2 mole of Q would be: (a) 140 torr (b) 20 torr (c) 68 torr (d) 72 torr

    		•
  • Q : Describe Thermodynamics Properties The

    The free energy property leads to convenient expressions for the volume and pressure dependence of internal energy, enthalpy and the heat capacities.All the properties of a chemical system, a sample of a substance, or a mixture of substances have some fixe

    		•
  • Q : Colligative properties give atleast two

    give atleast two application of following colligative properties

    		•
  • Q : Describe the properties of the

    Briefly describe the properties of the carbohydrates?

    		•
  • Q : Strength of any solution Give me answer

    Give me answer of this question. A solution contains 1.2046 x 1024 hydrochloric acid molecules in one dm3 of the solution. The strength of the solution is: (a) 6 N (b) 2 N (c) 4 N (d) 8 N

    		•
  • Q : Moles of chloride ion Select the right

    Select the right answer of the question. A solution of CaCl2 is 0.5 mol litre , then the moles of chloride ion in 500ml will be : (a) 0.25 (b) 0.50 (c) 0.75 (d)1.00

    		•
  • Q : Film Mass Transport Sulfur trioxide

    Sulfur trioxide (SO3) is manufactured by the gas-phase oxidation of SO2 over a platinum catalyst: SO2 + ½ O2 à SO3 The catalyst is a non-porous ext

    		•
  • Q : Problem on thermodynamic equilibrium In

    In the manufacture of sulphuric acid by the contact process, S02 is oxidized to SO3 over a vanadium catalyst: The reactor is adiabatic and operates at atmospheric pressure. The gases enter the reactor at 410&d

    		•
  • Q : Explain solid in liquid solutions. The

    The French chemist Francois Marie Raoult (1886) carried out a series of experiments to study the vapour pressure of a number of binary solutions. On the basis of the results of the experiments, he proposed a generalization called Raoult's law which states that, <

    		•