# How many quantum numbers are needed to define the probability of finding the electron in a given region of space in the hydrogen atom?

##### 1 Answer

Three:

We need the quantum level of the orbital, the shape/type (

**PROBABILITY DENSITY**

*The probability density of an electron in an atomic orbital is defined as the statistical distribution of where an electron often appears in that orbital.*

**Probability density** is defined as:

#int_"allspace" psi^"*"psid tau# where:

#psi# is thewave functionthat describes the state of a quantum mechanical system such as an atom or molecule.#psi^"*"# is itscomplex conjugate; since Real Chemistsâ„˘ use real numbers,#psi^"*" = psi# .- "allspace" means the inclusion of all possible (finite) locations in space.

**PROBABILITY DENSITY FOR THE HYDROGEN ATOM**

For the *hydrogen atom*, which uses **spherical coordinates** (**radial** and **angular** components,

#\mathbf(psi(vecr,theta,phi) = R_(nl)(vecr)Y_(l)^(m_l)(theta,phi))#

From this, the probability density is written as:

#int_"allspace" psi^"*"(vecr,theta,phi)psi(vecr,theta,phi) d tau = stackrel("Radial Component")overbrace(int_(0)^(oo) R_(nl)^2(vecr)r^2dr) stackrel("Angular Component")overbrace(int_(0)^(pi) (Y_(l)^(m_l)(theta))^2sinthetad theta int_(0)^(2pi) (Y_(l)^(m_l)(phi))^2dphi)#

- The
radial componentcontains the quantum numbers#n# (quantum level, such as#n = color(blue)(3)# for the#color(blue)(3)s# atomic orbital, etc) and#l# (theangular momentum, which for instance is#l = color(blue)(2)# for a#color(blue)(d)# orbital).#vecr# is theradial distance(outwards in all directions).- The
angular componentessentiallysweeps out the changes awayfromspherical uniformity. For instance, an#s# orbital has a constant#theta# and#phi# , so it is a sphere. A#p# orbital is not a sphere, so it has#theta# and#phi# dependence.#theta# is theangle from the#\mathbf(z)# axis(vertical) going clockwise towards the#x# axis (horizontal), and#phi# is theangle from the#\mathbf(y)# axis(towards you) going counterclockwise (on the#xy# -plane).

**WHAT DO WE PLOT?**

*For simplicity, we often don't plot* *We choose to plot the radial density distribution by using only the radial component.*

Therefore, it is a graph of

**TAKE-HOME MESSAGE**

Since

#n# to show the quantum level of the orbital that the electron might be found in.#l# to show the shape of the orbital that the electron might be found in.#m_l# to determine which of the#2l + 1# orbitals (#m_l = 0, pm1, . . . , pml# ) the electron might be found in.