Quantum Mechanical Model

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JEE Advanced 2026

The 2s and the 2p orbital energies of hydrogen atom are $E_{2 s} (H)$ and $E_{2 p} (H)$ , respectively. The 2s and the 2p orbital energies of lithium atom are $E_{2 s} (Li)$ and $E_{2 p} (Li)$ , respectively. The correct option (s) about the orbital energies is (are)

E_{2 s} (Li) < E_{2 p} (Li)

(B)

E_{2 s} (H) = E_{2 p} (H)

(C)

E_{2 p} (H) < E_{2 s} (Li)

(D)

E_{2 s} (H) > E_{2 s} (Li)

JEE Advanced 2016

LEVELJEE Main

P is the probability of finding the 1s electron of hydrogen atom in a spherical shell of infinitesimal thickness, dr, at a distance r from the nucleus. The volume of this shell is $4 π r^{2} dr$ . The qualitative sketch of the dependence of P on r is -

Comprehension Passage

The wave function

Ψ_{n, l, m_{l}}

is a mathematical function whose value depends upon spherical polar coordinates

(r, θ, ϕ)

of the electron and characterized by the quantum numbers

n

l

and

m_{l}

. Here

r

is distance from nucleus,

θ

is colatitude and

ϕ

is azimuth. In the mathematical functions given in the Table,

Z

is atomic number

a_{0}

is Bohr radius.

Column-1 (I) 1s orbital (II) 2s orbital (III) 2 p_{z} orbital (IV) 3 d_{z^{2}} orbital Column-2 (i) Ψ_{n, l, m_{l}} \propto (\frac{Z}{a _{0}})^{\frac{3}{2}} e^{- (\frac{Z r}{a _{0}})} (ii) One radial node (iii) Ψ_{n, l, m_{l}} \propto (\frac{Z}{a _{0}})^{\frac{5}{2}} r e^{- (\frac{Z r}{2 a _{0}})} cos θ (iv) xy - plane is a nodal plane Column-3 (P) [See Graph in Image] (Q) Probability density at nucleus \propto \frac{1}{a _{0}^{3}} (R) Probability density is maximum at nucleus (S) Energy needed to excite electron from n = 2 state to n = 4 state is \frac{27}{32} times the energy needed to excite electron from n = 2 state to n = 6 state

Question 1:

For the given orbital in column 1, the only CORRECT combination for any hydrogen - like species is :

For $He^{+}$ ion, the only INCORRECT combination is

For hydrogen atom, the only CORRECT combination is

The ground state energy of hydrogen atom is $- 13.6 eV$ . Consider an electronic state $Ψ$ of $He^{+}$ whose energy, azimuthal quantum number and magnetic quantum number are $- 3.4 eV$ , $2$ and $0$ respectively. Which of the following statement(s) is(are) true for the state $Ψ$ ?

* Multiple Correct Options

(A)

It has

2

angular nodes

(B)

It has

3

radial nodes

(C)

It is a

4 d

state

(D)

The nuclear charge experienced by the electron in this state is less than

2 e

, where

e

is the magnitude of the electronic charge.

JEE Advanced 2015

LEVELJEE Main

Not considering the electronic spin the degeneracy of the second excited state ( $n = 3$ ) of H-atom is 9, where the degeneracy of the second excited state of $H^{-}$ is

JEE Advanced 2014

LEVELJEE Main

In an atom, the total number of electrons having quantum numbers $n = 4$ , $∣ m_{l} ∣ = 1$ and $m_{s} = - \frac{1}{2}$ is

JEE Main 2019

LEVELJEE Main

The graph between $∣ ψ ∣^{2}$ and $r$ (radial distance) is shown below. This represents

The quantum number of four electrons are given below: I. $n = 4, l = 2, m_{l} = - 2, m_{s} = - \frac{1}{2}$ II. $n = 3, l = 2, m_{l} = 1, m_{s} = + \frac{1}{2}$ III. $n = 4, l = 1, m_{l} = 0, m_{s} = + \frac{1}{2}$ IV. $n = 3, l = 1, m_{l} = 1, m_{s} = - \frac{1}{2}$ The correct order of their increasing energies will be

Consider the hypothetical situation where the azimuthal quantum number, $l$ takes values $0, 1, 2, \dots n + 1$ , where $n$ is the principle quantum number. Then, the element with atomic number

(A)

8 is the first noble gas

(B)

13 has a half-filled valence subshell

(C)

9 is the first alkali metal

(D)

6 has a 2p-valence subshell

JEE Main 2019

LEVELJEE Main

Among the following, the energy of 2s-orbital is lowest in

The number of orbitals with $n = 5$ , $m_{l} = + 2$ is ......... . (Round off to the nearest integer).

JEE Main 2019

LEVELJEE Advanced

Which of the following combination of statements is true regarding the interpretation of the atomic orbitals? I. An electron in an orbital of high angular momentum stays away from the nucleus than an electron in the orbital of lower angular momentum. II. For a given value of the principal quantum number, the size of the orbit is inversely proportional to the azimuthal quantum number. III. According to wave mechanics, the ground state angular momentum is equal to $\frac{h}{2 π}$ . IV. The plot of $ψ$ vs $r$ for various azimuthal quantum numbers, shows peak shifting towards higher $r$ value.

The value of magnetic quantum number of the outermost electron of $Zn^{+}$ ion is ......... .

JEE Main 2021

LEVELJEE Main

The plots of radial distribution functions for various orbitals of hydrogen atom against 'r' are given below. The correct plot for 3s-orbital is

JEE Main 2020, 5 Sep Shift-II

LEVELJEE Main

The correct statement about probability density (except at infinite distance from nucleus) is

(A)

it can be zero for 1s orbital

(B)

it can be negative for 2p orbital

(C)

it can be zero for 3p orbital

(D)

it can never be zero for 2s orbital

JEE Main 2020

LEVELJEE Main

The number of subshells associated with $n = 4$ and $m = - 2$ quantum numbers is

A certain orbital has $n = 4$ and $m_{l} = - 3$ . The number of radial nodes in this orbital is ...... (Round off to the nearest integer).

JEE Main 2021

LEVELJEE Main

The spin only magnetic moment of a divalent ion in aqueous solution (atomic number = 29) is ……… BM.

JEE Main 2019

LEVELJEE Main

Which one of the following about an electron occupying the $1 s$ -orbital in a hydrogen atom is incorrect? (The Bohr radius is represented by $a_{0}$ )

(A)

The electron can be found at a distance

2 a_{0}

from the nucleus.

(B)

The magnitude of the potential energy is double that of its kinetic energy on an average.

(C)

The probability density of finding the electron is maximum at the nucleus.

(D)

The total energy of the electron is maximum when it is at a distance

a_{0}

from the nucleus.

JEE Main 2020

LEVELJEE Main

The number of orbitals associated with quantum numbers $n = 5$ , $m_{s} = + \frac{1}{2}$ is

The correct set of four quantum numbers for the valence electrons of rubidium atom $[Z = 37]$ is

(A)

5, 0, 0, + \frac{1}{2}

(B)

5, 1, 0, + \frac{1}{2}

(C)

5, 1, 1, + \frac{1}{2}

(D)

5, 0, 1, + \frac{1}{2}

Comprehension Passage

For the given orbital in column 1, the only CORRECT combination for any hydrogen - like species is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For He+ ion, the only INCORRECT combination is

For hydrogen atom, the only CORRECT combination is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Not considering the electronic spin the degeneracy of the second excited state (n=3) of H-atom is 9, where the degeneracy of the second excited state of H− is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In an atom, the total number of electrons having quantum numbers n=4, ∣ml​∣=1 and ms​=−21​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The graph between ∣ψ∣2 and r (radial distance) is shown below. This represents

The electrons are more likely to be found

The orbital having two radial as well as two angular nodes is

A certain orbital has no angular nodes and two radial nodes. The orbital is

Among the following, the energy of 2s-orbital is lowest in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of orbitals with n=5, ml​=+2 is ......... . (Round off to the nearest integer).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of magnetic quantum number of the outermost electron of Zn+ ion is ......... .

The plots of radial distribution functions for various orbitals of hydrogen atom against 'r' are given below. The correct plot for 3s-orbital is

The correct statement about probability density (except at infinite distance from nucleus) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of subshells associated with n=4 and m=−2 quantum numbers is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A certain orbital has n=4 and ml​=−3. The number of radial nodes in this orbital is ...... (Round off to the nearest integer).

The spin only magnetic moment of a divalent ion in aqueous solution (atomic number = 29) is ……… BM.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which one of the following about an electron occupying the 1s-orbital in a hydrogen atom is incorrect? (The Bohr radius is represented by a0​)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of orbitals associated with quantum numbers n=5, ms​=+21​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The correct set of four quantum numbers for the valence electrons of rubidium atom [Z=37] is

For $He^{+}$ ion, the only INCORRECT combination is

Not considering the electronic spin the degeneracy of the second excited state ( $n = 3$ ) of H-atom is 9, where the degeneracy of the second excited state of $H^{-}$ is

In an atom, the total number of electrons having quantum numbers $n = 4$ , $∣ m_{l} ∣ = 1$ and $m_{s} = - \frac{1}{2}$ is

The graph between $∣ ψ ∣^{2}$ and $r$ (radial distance) is shown below. This represents

The number of orbitals with $n = 5$ , $m_{l} = + 2$ is ......... . (Round off to the nearest integer).

The value of magnetic quantum number of the outermost electron of $Zn^{+}$ ion is ......... .

The number of subshells associated with $n = 4$ and $m = - 2$ quantum numbers is

A certain orbital has $n = 4$ and $m_{l} = - 3$ . The number of radial nodes in this orbital is ...... (Round off to the nearest integer).

Which one of the following about an electron occupying the $1 s$ -orbital in a hydrogen atom is incorrect? (The Bohr radius is represented by $a_{0}$ )

The number of orbitals associated with quantum numbers $n = 5$ , $m_{s} = + \frac{1}{2}$ is

The correct set of four quantum numbers for the valence electrons of rubidium atom $[Z = 37]$ is