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Question: In \({\psi _{321}}\) the sum of angular momentum, spherical nodes and angular nodes is. A) \(\dfra...

In ψ321{\psi _{321}} the sum of angular momentum, spherical nodes and angular nodes is.
A) 6h+4π2π\dfrac{{\sqrt {6h} + 4\pi }}{{2\pi }}
B) 6h2π+3\dfrac{{\sqrt {6h} }}{{2\pi }} + 3
C) 6h+2π2π\dfrac{{\sqrt {6h} + 2\pi }}{{2\pi }}
D) 6h+8π2π\dfrac{{\sqrt {6h} + 8\pi }}{{2\pi }}

Explanation

Solution

We can calculate the angular momentum using the formula,
Angular momentum=l(l+1).h2π = \sqrt {l\left( {l + 1} \right)} .\dfrac{h}{{2\pi }}
Where h is the Planck constant.
The number of spherical nodes in an orbit can be calculated by n−l−1n - l - 1 where n is the principal quantum number and l is the angular momentum quantum number.

Complete step by step answer:
The wave function ψ\psi represents an orbital. In orbitalψ321{\psi _{321}}, 3 represents principal quantum number n, 2 represents the angular momentum quantum number l and 1 represents m.
Now, calculate the angular momentum,
Angular momentum=l(l+1).h2π = \sqrt {l\left( {l + 1} \right)} .\dfrac{h}{{2\pi }}
Angular momentum=2(2+1).h2π=6h2π = \sqrt {2\left( {2 + 1} \right)} .\dfrac{h}{{2\pi }} = \dfrac{{\sqrt 6 h}}{{2\pi }}
The number of spherical nodes in an orbit can be calculated as,
Radical nodes=n−l−1=3−2−1=0 = n - l - 1 = 3 - 2 - 1 = 0
The angular node l=2l = 2
Thus the sum of angular momentum and the nodes is 6.h2π+2=6.h+4π2π\dfrac{{\sqrt 6 .h}}{{2\pi }} + 2 = \dfrac{{\sqrt 6 .h + 4\pi }}{{2\pi }}

So, the correct answer is Option A.

Additional Information:
We must remember that the wave function helps us to define the probability of the quantum state of an element as a function of position, momentum, time and spin. It is represented by Greek alphabet psi ψ\psi .
Though, it is important to note that there is no physical importance of wave function itself. Nevertheless its proportionate value of ψ2{\psi _2} at a given time and point of space does have physical importance.

Note:
Let us discuss the properties of wave function.
Properties of Wave Function:
Wave function Ψ\Psi contains all the measurable information about the particles.∫−∞+∞ψ2dxdydz=1\int_{ - \infty }^{ + \infty } {{\psi ^2}dxdydz = 1} This covers all possibilities.
The wave function Ψ\Psi is finite and it is continuous because the wave function should be able to describe the behaviour of a particle across all potentials, in any region.
The wave function Ψ\Psi must be single valued. This is a way of guaranteeing that there is only a single value for the probability of the system being in a given state.
Hence, all the given options are correct.
The linear partial equation describing the wave function is called the Schrodinger equation. The equation is named after Schrodinger.
The equations of Schrodinger equation are,
Time dependent Schrodinger equation: ih∂∂tΨ(r,t)=h22m▽2+V(r,t)]Ψ(r,t)ih\partial \partial t\Psi \left( {r,t} \right) = h22m\triangledown 2 + V\left( {r,t} \right)]\Psi \left( {r,t} \right)
Time independent Schrodinger equation: −h22m▽2+V(r)]Ψ(r)=EΨ(r)- h22m\triangledown 2 + V\left( r \right)]\Psi \left( r \right) = E\Psi \left( r \right).