Classical mechanics, based on Newton’s laws of motion, successfully describes the motion of all macroscopic objects such as a falling stone, orbiting planets etc., which have essentially a particle-like behaviour as shown in the previous section. However it fails when applied to microscopic objects like electrons, atoms, molecules etc. This is mainly because of the fact that classical mechanics ignores the concept of dual behaviour of matter especially for sub-atomic particles and the uncertainty principle.
-- NCERT Class 11 Chemistry, Ch. 2, p. 53Quantum Mechanical Model
8 MCQs9-step worked example
Source: NCERT Structure of AtomOfficial key: NTA-verifiedLast updated: 8 Oct 2026
Try this first
Which branch of science takes into account the dual (wave and particle) behaviour of matter for sub-atomic particles?
- A.Classical mechanics
- B.Quantum mechanics
- C.Thermodynamics
- D.Kinetic theory of gases
Tap to see the answer
Answer: B. Quantum mechanics is the branch of science that takes the dual behaviour of matter into account (NCERT Class 11 Chemistry, Chapter 2, page 53).
A is wrong: A is wrong because classical mechanics ignores the dual behaviour of matter and the uncertainty principle, which is why it fails for electrons (NCERT Class 11 Chemistry, Chapter 2, page 53).
C is wrong: C is wrong because thermodynamics deals with energy changes of macroscopic systems, not with the wave nature of electrons (trap: picking a familiar chapter name).
D is wrong: D is wrong because the kinetic theory treats gas molecules as particles and does not use the wave behaviour of matter (trap: picking a familiar chapter name).
Quantum Mechanical Model, explained for NEET
Start with a question: if nobody can ever say exactly where an electron is, how can chemistry still predict the energies and shapes of atoms so well? The quantum mechanical model answers by changing what we ask. It stops asking for a path and asks for a probability.
Classical mechanics, built on Newton's laws, describes macroscopic objects such as a falling stone or an orbiting planet. It fails for electrons, atoms and molecules mainly because it ignores the dual behaviour of matter and the uncertainty principle (NCERT Class 11 Chemistry, Chapter 2, page 53).
Quantum mechanics is the branch of science that takes this dual behaviour into account. When it is applied to macroscopic objects, whose wave-like properties are insignificant, it gives the same results as classical mechanics (NCERT Class 11 Chemistry, Chapter 2, page 53).
The fundamental equation was developed by Schrödinger. Its Hamiltonian is built from the total energy of the system, and solving the equation gives E and ψ (NCERT Class 11 Chemistry, Chapter 2, page 53). For the hydrogen atom the solution gives the allowed energy levels and the wave function for each, characterised by three quantum numbers: n, l and ml. These arise as a natural consequence of solving the equation (NCERT Class 11 Chemistry, Chapter 2, page 53).
The wave function ψ is a mathematical function of the electron's coordinates and carries no physical meaning. The wave functions of one-electron species are called atomic orbitals (NCERT Class 11 Chemistry, Chapter 2, page 54). What does carry meaning is |ψ|²: the probability of finding the electron at a point is proportional to |ψ|² there, and |ψ|² is called the probability density, always positive (NCERT Class 11 Chemistry, Chapter 2, page 54).
The model predicts all aspects of the hydrogen spectrum, including phenomena Bohr's model could not explain (NCERT Class 11 Chemistry, Chapter 2, page 54). For multi-electron atoms the equation cannot be solved exactly, so approximate methods are used. The orbitals are somewhat contracted compared with hydrogen, and their energies depend on both n and l, whereas for hydrogen the energy depends only on n (NCERT Class 11 Chemistry, Chapter 2, page 54).
Key features of the model (NCERT Class 11 Chemistry, Chapter 2, page 54):
- Electron energy is quantized, a direct result of the wave-like properties of electrons.
- Exact position and exact velocity cannot both be known, so only the probability of finding the electron at different points is discussed.
- An atomic orbital is the wave function ψ for an electron; each orbital gives the electron a definite energy and cannot contain more than two electrons.
Watch out: ψ is not a probability. The probability is tied to |ψ|², so a point where ψ is negative is no less likely than one where ψ is positive of the same size.
How do you solve a Quantum Mechanical Model question? A worked example
- 1
Given
At two points A and B near the nucleus of an atom, the orbital wave function has the values ψA = 0.30 and ψB = −0.40, in the same arbitrary units.
- 2
Required
Which point is the electron more likely to be found at, and by what factor.
- 3
Concept
The probability of finding the electron at a point is proportional to |ψ|² at that point; ψ itself has no physical meaning (NCERT Class 11 Chemistry, Chapter 2, page 54).
- 4
Formula
Probability density ∝ |ψ|²
- 5
Substitution
At A: |ψA|² = (0.30)². At B: |ψB|² = (−0.40)².
- 6
Calculation
|ψA|² = 0.090 and |ψB|² = 0.16. The ratio is 0.16 / 0.090 = 1.78. The inputs have two significant figures, so the ratio is 1.8.
- 7
Final answer
The electron is about 1.8 times more likely to be found at B than at A, even though ψB is negative.
- 8
Common trap
Comparing ψ directly (0.30 > −0.40) points to A. That treats ψ as a probability, which it is not.
- 9
Similar NEET-style question
At two points the wave function has the values +0.50 and −0.50. How do the probabilities compare? (Answer: |ψ|² = 0.50 × 0.50 = 0.25 at the first point, and (−0.50) × (−0.50) = 0.25 at the second, so the two points are equally likely.)
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Can you answer these Quantum Mechanical Model MCQs?
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
MCQ 1Easy RecallPractice
Which branch of science takes into account the dual (wave and particle) behaviour of matter for sub-atomic particles?
Show answer and why every option is right or wrong
Answer: B. Quantum mechanics is the branch of science that takes the dual behaviour of matter into account (NCERT Class 11 Chemistry, Chapter 2, page 53).
Why A is wrong: A is wrong because classical mechanics ignores the dual behaviour of matter and the uncertainty principle, which is why it fails for electrons (NCERT Class 11 Chemistry, Chapter 2, page 53).
Why C is wrong: C is wrong because thermodynamics deals with energy changes of macroscopic systems, not with the wave nature of electrons (trap: picking a familiar chapter name).
Why D is wrong: D is wrong because the kinetic theory treats gas molecules as particles and does not use the wave behaviour of matter (trap: picking a familiar chapter name).
MCQ 2Easy RecallPractice
According to the quantum mechanical model, the probability of finding an electron at a point within an atom is proportional to
Show answer and why every option is right or wrong
Answer: B. The probability of finding an electron at a point is proportional to |ψ|² at that point, and |ψ|² is known as the probability density (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why A is wrong: A is wrong because ψ itself is a mathematical function with no physical meaning and can even be negative; only its square relates to probability (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why C is wrong: C is wrong because the probability rises with |ψ|², not with its reciprocal; a large |ψ|² means a likely region (trap: inverting the relation).
Why D is wrong: D is wrong because E is the energy of the state, obtained together with ψ from the Schrödinger equation, and does not give the probability at a point (NCERT Class 11 Chemistry, Chapter 2, page 53).
MCQ 3Easy RecallPractice
In the quantum mechanical model, an atomic orbital is
Show answer and why every option is right or wrong
Answer: B. An atomic orbital is the wave function ψ for an electron in an atom; when an electron is described by a wave function, it is said to occupy that orbital (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why A is wrong: A is wrong because the path of an electron in an atom can never be determined or known accurately (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why C is wrong: C is wrong because the model only gives the region where the electron will most probably be found, never a certain location (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why D is wrong: D is wrong because an orbital is a wave function in which the electron has a definite energy; it is not the same thing as an energy level (NCERT Class 11 Chemistry, Chapter 2, page 54).
MCQ 4Direct ApplicationPractice
Which of the following can quantum mechanics give for an electron in an atom?
Show answer and why every option is right or wrong
Answer: C. Exact position and exact velocity cannot be determined together, so one talks only of the probability of finding the electron at different points, which follows from |ψ|² (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why A is wrong: A is wrong because both the exact position and exact velocity of an electron cannot be determined simultaneously (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why B is wrong: B is wrong because the path of an electron in an atom can never be determined or known accurately (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why D is wrong: D is wrong because the model gives no exact position at all, only a probability; it does not trade away velocity to recover a position (trap: reading the uncertainty principle as a choice between the two).
MCQ 5Direct ApplicationPractice
Compare the 2s and 2p orbitals. Which statement is correct?
Show answer and why every option is right or wrong
Answer: C. In hydrogen or hydrogen-like species the orbital energy depends only on n, so 2s and 2p match. In multi-electron atoms the energies depend on n and l, so they differ (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why A is wrong: A is wrong because in multi-electron atoms the energy depends on l as well as n, so orbitals with the same n but different l separate (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why B is wrong: B is wrong because in hydrogen the energy depends only on n, so 2s and 2p are equal there (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why D is wrong: D is wrong because it reverses the two cases; dependence on l appears only for multi-electron atoms (trap: swapping hydrogen and multi-electron behaviour).
MCQ 6Direct ApplicationPractice
When the Schrödinger equation is solved for the hydrogen atom, the quantized energy states and wave functions are characterised by which set of quantum numbers?
Show answer and why every option is right or wrong
Answer: B. The page names the principal quantum number n, the azimuthal quantum number l and the magnetic quantum number ml as the three that arise naturally from the solution (NCERT Class 11 Chemistry, Chapter 2, page 53).
Why A is wrong: A is wrong because the spin quantum number s is not among the three that the page says arise from solving the equation for hydrogen (NCERT Class 11 Chemistry, Chapter 2, page 53).
Why C is wrong: C is wrong because it drops l and includes s; the three named on the page are n, l and ml (NCERT Class 11 Chemistry, Chapter 2, page 53).
Why D is wrong: D is wrong because it leaves out n, which fixes the energy of hydrogen's orbitals (NCERT Class 11 Chemistry, Chapter 2, page 54).
MCQ 7CalculationPractice
At a point in an atom the wave function of an electron has the value ψ = −0.20 (arbitrary units). Which statement is correct?
Show answer and why every option is right or wrong
Answer: B. Squaring gives |ψ|² = (−0.20)² = 0.040. The probability density is always positive, and the probability is proportional to it, so a negative ψ does not rule the point out (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why A is wrong: A is wrong because probability follows |ψ|², which is always positive; ψ itself has no physical meaning (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why C is wrong: C is wrong because ψ is not a probability; the probability is proportional to |ψ|² = 0.040 (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why D is wrong: D is wrong because the sign of ψ says nothing about whether the electron can be found at the point; |ψ|² is what matters (trap: reading the sign of ψ as presence or absence).
MCQ 8CalculationPractice
Consider these statements about the quantum mechanical model. P: the energy of electrons in atoms is quantized. Q: both the exact position and exact velocity of an electron can be determined simultaneously. R: quantized energy levels are a direct result of the wave-like properties of electrons. S: an orbital can contain three electrons. Which statements are correct?
Show answer and why every option is right or wrong
Answer: A. P and R are two of the model's stated features: energy is quantized, and the quantized levels follow from the wave-like properties of electrons. Q contradicts the uncertainty principle, and S contradicts the limit of two electrons per orbital (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why B is wrong: B is wrong because Q is false: exact position and exact velocity cannot be determined simultaneously (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why C is wrong: C is wrong because both are false: Q breaks the uncertainty principle and an orbital cannot contain more than two electrons (NCERT Class 11 Chemistry, Chapter 2, page 54).
Why D is wrong: D is wrong because S is false: an orbital cannot contain more than two electrons (NCERT Class 11 Chemistry, Chapter 2, page 54).
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What to remember before solving Quantum Mechanical Model questions
11 NCERT lines
The branch of science that takes into account this dual behaviour of matter is called quantum mechanics. Quantum mechanics is a theoretical science that deals with the study of the motions of the microscopic objects that have both observable wave like and particle like properties. It specifies the laws of motion that these objects obey. When quantum mechanics is applied to macroscopic objects (for which wave like properties are insignificant) the results are the same as those from the classical mechanics.
-- NCERT Class 11 Chemistry, Ch. 2, p. 53The total energy of the system takes into account the kinetic energies of all the sub-atomic particles (electrons, nuclei), attractive potential between the electrons and nuclei and repulsive potential among the electrons and nuclei individually. Solution of this equation gives E and ψ.
-- NCERT Class 11 Chemistry, Ch. 2, p. 53When Schrödinger equation is solved for hydrogen atom, the solution gives the possible energy levels the electron can occupy and the corresponding wave function(s) (ψ) of the electron associated with each energy level. These quantized energy states and corresponding wave functions which are characterized by a set of three quantum numbers (principal quantum number n, azimuthal quantum number l and magnetic quantum number ml ) arise as a natural consequence in the solution of the Schrödinger equation.
-- NCERT Class 11 Chemistry, Ch. 2, p. 53The wave function is a mathematical function whose value depends upon the coordinates of the electron in the atom and does not carry any physical meaning. Such wave functions of hydrogen or hydrogen like species with one electron are called atomic orbitals.
-- NCERT Class 11 Chemistry, Ch. 2, p. 54Key Fact
Probability of finding the electron is proportional to |ψ|²; QM explains the whole hydrogen spectrum
The probability of finding an electron at a point within an atom is proportional to the |ψ|2 at that point. The quantum mechanical results of the hydrogen atom successfully predict all aspects of the hydrogen atom spectrum including some phenomena that could not be explained by the Bohr model.
-- NCERT Class 11 Chemistry, Ch. 2, p. 54Key Fact
Multi-electron atoms: equation not solvable exactly; orbitals contracted; energies depend on n and l
Application of Schrödinger equation to multi-electron atoms presents a difficulty: the Schrödinger equation cannot be solved exactly for a multi-electron atom. This difficulty can be overcome by using approximate methods. Such calculations with the aid of modern computers show that orbitals in atoms other than hydrogen do not differ in any radical way from the hydrogen orbitals discussed above. The principal difference lies in the consequence of increased nuclear charge. Because of this all the orbitals are somewhat contracted. Further, as you shall see later (in subsections 2.6.3 and 2.6.4), unlike orbitals of hydrogen or hydrogen like species, whose energies depend only on the quantum number n, the energies of the orbitals in multi-electron atoms depend on quantum numbers n and l.
-- NCERT Class 11 Chemistry, Ch. 2, p. 54Key Fact
Features 1-2: electron energy is quantized, a direct result of the wave-like properties of electrons
Quantum mechanical model of atom is the picture of the structure of the atom, which emerges from the application of the Schrödinger equation to atoms. The following are the important features of the quantum-mechanical model of atom: 1. The energy of electrons in atoms is quantized (i.e., can only have certain specific values), for example when electrons are bound to the nucleus in atoms. 2. The existence of quantised electronic energy levels is a direct result of the wave like properties of electrons and are allowed solutions of Schrödinger wave equation.
-- NCERT Class 11 Chemistry, Ch. 2, p. 54Key Fact
Feature 3: exact position and velocity cannot be known together, so only probabilities are spoken of
Both the exact position and exact velocity of an electron in an atom cannot be determined simultaneously (Heisenberg uncertainty principle). The path of an electron in an atom therefore, can never be determined or known accurately. That is why, as you shall see later on, one talks of only probability of finding the electron at different points in an atom.
-- NCERT Class 11 Chemistry, Ch. 2, p. 54An atomic orbital is the wave function ψ for an electron in an atom. Whenever an electron is described by a wave function, we say that the electron occupies that orbital. Since many such wave functions are possible for an electron, there are many atomic orbitals in an atom. These “one electron orbital wave functions” or orbitals form the basis of the electronic structure of atoms. In each orbital, the electron has a definite energy. An orbital cannot contain more than two electrons.
-- NCERT Class 11 Chemistry, Ch. 2, p. 54The probability of finding an electron at a point within an atom is proportional to the square of the orbital wave function i.e., |ψ|2 at that point. |ψ|2 is known as probability density and is always positive. From the value of |ψ|2 at different points within an atom, it is possible to predict the region around the nucleus where electron will most probably be found.
-- NCERT Class 11 Chemistry, Ch. 2, p. 54Quantum Mechanical Model: NEET previous year questions (PYQs) with answers
9 questions in Structure of Atom
No question in our NEET 2020–2025 set targets this topic directly.
More in Structure of Atom: 4 exam traps and mistakes · 5 formulas · 3 question patterns from its other lessons.
Sources
NCERT refs: Class 11 Chemistry Chapter 2, p.53 | Class 11 Chemistry Chapter 2, p.54
Page numbers are the ones printed in the current NCERT textbook (2023 rationalised edition), unless marked pre-2023. The books are free at ncert.nic.in.
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