Particle In Three Dimensional Box - Chemistry Aniket Ncert

What is a Wave function in quantum physics?

A wave function is a mathematical description of a quantum state of a particle as a function of momentum, time, position and spin. The symbol used for a wave function is a Greek letter called psi, ψ. It helps us to know the probability of finding particle within the matter.

Characteristics of wave function :- 

1. Contains all measurable information about the particle.
2. ψ should be continuous and single-valued.
3. ψ should be finite.
4. ψ should be square Integrable.
5. ψ2 gives us probability of finding the particle.

Boundary Conditions: -

• In order to avoid infinite probabilities, ψ must be finite everywhere.
• In order to avoid multiple values of the probability, ψ must be single valued.
• For finite potentials, the wave function and its derivative must be continuous. This is required because the second order derivative term in the wave equation must be single valued.
• In order to normalize the wave function, it must approach zero as x approaches infinity.

Particle in a Three Dimensional Box :- 

The quantum particle in the 1D box problem can be expanded to consider a particle within higher dimensions.

Consider a particle which can move freely with in a box of dimensions a × b × c with impenetrable walls. The potential can be written mathematically as:- 

𝑉 = {0 𝑖𝑛𝑠𝑖𝑑𝑒 at surfaces and outside

Since the wavefunction ψ should be well behaved, so, it must vanish everywhere outside the box. By the continuity requirement, the wavefunction must also vanish in the six surfaces of the box. Orienting the box so its edges are parallel to the cartesian axes, with one corner at (0,0,0), the following boundary conditions must be satisfied:

ψ (x,y, z) = 0 when x = 0, x = a, y = 0, y = b, z = 0 or z = c

Inside the box, where the potential energy is l everywhere zero, the Hamiltonian is simply the 3-D kinetic energy operator and the Schrodinger equation reads

The easiest way in solving this partial differential equation is by having the wavefunction equal to a product of individual function for each independent variable (the Separation of Variables technique):

• X(x) is a function of variable x only
• Y(y) is a function of variable y only
• Z(z) is a function of variable z only

On substituting Eq. (3) into Eq.(2) and division by X(x)Y(y)Z(z) , we obtain 

Each of the first three terms depends on one variable only, independent of the other two. We can write Eq.(4) as;

Now on LHS of Eq. (5) we have only function of x, while right hand side (RHS) contains functions of y and z. This is possible only if each term separately equals a constant, say, −α

2.

Each of the three equations with its associated boundary conditions is equivalent to the one-dimensional problem. The normalized solutions X(x), Y (y), Z(z) can therefore be written down in complete analogy with

one dimensional box.

Quantum systems with symmetry generally exhibit degeneracy in their energy levels. This means that there can exist distinct eigenfunctions which share the same eigenvalue. An eigenvalue which corresponds to a unique eigenfunction is termed non degenerate while one which belongs to n different eigenfunctions is termed n-fold degenerate.

The ground state has only one wave function and no other state has this specific energy; the ground state and the energy level are said to be non-degenerate

However, in the 3-D cubical box potential the energy of a state depends upon the sum of the squares of the quantum numbers. The particle having a particular value of energy in the excited state MAY has several different stationary states or wavefunctions. If so, these states and energy eigenvalues are said to be degenerate.

For the first excited state, three combinations of the quantum numbers (nx,ny,nz) are (2,1,1),(1,2,1),(1,1,2)(2,1,1),(1,2,1),(1,1,2). The sum of squares of the quantum numbers in each combination is same (equal to 6). Each wave function has same energy:

Corresponding to these combinations three different wave functions

and three different states are possible. Hence, the first excited state is said to be three-

fold or triply degenerate. The number of independent wave functions for the stationary

states of an energy level is called as the degree of degeneracy of the energy level.