# Schrödinger's wave equation gives which type of concept of quantum number?

##### 1 Answer

The **Schrodinger equation** is constrained so that the energies that arise from solving it are all dependent in some way on quantum numbers, which are all integers (except for

**EXAMPLE 1: PARTICLE IN A 3D BOX**

For instance, for a system modeled as a three-dimensional **particle in a box**, the energy goes as

#E_n = (h^2)/(8m)(n_x^2/L_x^2 + n_y^2/L_y^2 + n_z^2/L_z^2)# ,

#n_q = 1, 2, 3, . . . # is theprincipal quantum numberfor the#q# th dimension, and everything else is a constant.

**EXAMPLE 2: HARMONIC OSCILLATOR**

For a system modeled as a **harmonic oscillator** (ball and spring), the energy goes as

#E_(upsilon) = hnu(upsilon + 1/2)# ,

#upsilon = 0, 1, 2, . . . # is thevibrational quantum number.#h# is Planck's constant in#"J"cdot"s"# and#nu# is the fundamental vibrational frequency in#"s"^(-1)# .

**EXAMPLE 3: DIATOMIC RIGID ROTATOR**

For a system modeled as a *diatomic***rigid rotator** (two balls connected by a stiff rod, rotating), the energy goes as

#E_J = J(J+1)B# ,

#J = 0, 1, 2, . . . # is therotational quantum numberfor a two-dimensional rigid rotator. Basically,#B# is the rotational constant, dependent on the identity of the molecule.

You can see that in all of these cases, in some way, the energy that arises from solving the Schrodinger equation is **restricted to certain energy levels** that are *directly tied* to the quantum number for that system.