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1.1 Postulates of Quantum Mechanics: Operators, Eigenfunctions, Hermitian Operators and Acceptable Wave Functions

★★★ 2021–2025: 11 Mains parts, 78 marks (Geo-Scientist 10 parts / 70 m · IFoS 1 / 8 m). Geo-Scientist Paper II asks one or two short parts every year (Q1(k), Q1(l)) and a 10–20 mark question on Hermitian operators: pure theory, quick to write, full marks if the steps are shown. Civil Services asks the same ideas inside particle-in-a-box and uncertainty questions.

Syllabus lines. Civil Services: “Heisenberg’s uncertainty principle, Schrödinger wave equation (time independent); interpretation of wave function”. IFoS: “Quantum theory, Heisenberg’s uncertainty principle, Schrödinger wave equation (time independent). Interpretation of wave function”. Geo-Scientist: “Eigenfunctions and eigenvalues. Uncertainty relation, expectation value. Hermitian operators. Schrödinger time-independent equation: nature of the equation, acceptability conditions imposed on the wave functions and probability interpretation of wave function.”

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CSE 2024 Q7(a) = asked there; tap to open the solved answerSN = line kept in the short notes

1.1.1 Postulates and the Born interpretation ★★★

  1. State. Everything that can be known about a system is in its wave function ψ(x, y, z, t). ψ itself is not observable; it may be negative or complex.SN
  2. Observables ↔ operators. Every observable (position, momentum, energy…) has a linear Hermitian operator: .SN
  3. Measurement. A single measurement of the observable gives only an eigenvalue a of its operator (Âψ = aψ). If ψ is an eigenfunction the result is certain; otherwise the average of many measurements is the expectation value (Eq. 1.1.8).SN
  4. Dynamics. ψ changes with time by the time-dependent Schrödinger equation ; for a state of fixed energy this separates into the time-independent equation Ĥψ = Eψ (Eq. 1.1.3).SN
(1.1.1)
  • Born interpretation: is the probability density; |ψ|² dx is the probability of finding the particle between x and x + dx. It is real and never negative even when ψ is negative or complex Geo 2024 Q1(l) 5m.SN
  • Dimensions: the integral of |ψ|² dx is a pure number and dx is a length, so in one dimension |ψ|² has units of m⁻¹ and ψ of m⁻¹ᐟ²; in three dimensions |ψ|² has m⁻³ (a probability per unit volume) Geo 2024 Q1(l).SN
xψ(x)+ lobe− lobeψ may be negative or complex: no meaning alonex|ψ|2 x1 x2 area = probabilitybetween x1 and x2|ψ|2 ≥ 0 everywhere; total area = 1node
Fig. 1.1.1 ψ can have either sign; ψ² cannot. The probability of finding the particle between x₁ and x₂ is the area under ψ² there.
Draw it in 60 s: draw ψ with a + lobe and a − lobe crossing zero at the origin; beside it ψ2 with two positive humps and a node at the origin; shade a thin strip and write “area = probability”.

1.1.2 Acceptable (well-behaved) wave functions ★★★

Because |ψ|² dx is a probability and ψ must satisfy a second-order differential equation, an acceptable ψ must be Geo 2025 Q1(k) 5m Geo 2021 Q1(k) 5m:SN

  1. Single-valued: one value at each point, otherwise the probability there is not unique. On a ring this forces , which quantises m.SN
  2. Finite everywhere (an infinite value over any finite region would give an infinite probability).SN
  3. Continuous, with a continuous first derivative: d²ψ/dx² in the Schrödinger equation must exist. The slope may jump only where V jumps to infinity, as at the walls of a box IFoS 2022 Q1(a) 8m.SN
  4. Square-integrable: ∫|ψ|² dx must be finite, so ψ → 0 as x → ±∞ and ψ can be normalised. A ψ that is zero everywhere is also rejected: the particle must be somewhere.SN
✓ acceptablee−x2 ✗ two values at one xnot single-valued✗ infinite at a point1/(4 − x)x = 4✗ jump in ψnot continuous✗ kink: slope jumpsallowed only at V = ∞✗ area of ψ2 infiniteex , −∞ < x < ∞
Fig. 1.1.2 One acceptable function and the five ways a function fails.
Draw it in 60 s: six small boxes: a bell (✓); a loop cut twice by a vertical line; a curve shooting up at x = 4; a step; a sharp V-tip; an exponential running off the top.

Solved PYQ IFoS 2022 Q1(a) 8m: why must φ and dφ/dx be continuous?

  1. Rewrite the Schrödinger equation: . If V and φ are finite, the right side is finite, so the second derivative is finite.
  2. Integrate across a point x₀ from x₀ − ε to x₀ + ε: as ε → 0. Hence dφ/dx is continuous, even at a finite step in V.SN
  3. If φ itself jumped, dφ/dx would be infinite there and d²φ/dx² undefined: no solution. Physically, a jump would make the probability density two-valued at that point. Only at an infinite wall (box) may the slope change abruptly.

Solved PYQ Geo 2023 Q7(b) 5m Geo 2021 Q1(k) 5m: allowed or forbidden?

  1. 1/(10x), 0 ≤ x ≤ ∞: infinite at x = 0 and ∫dx/100x² diverges there: forbidden.
  2. e−kx, −∞ ≤ x ≤ ∞ (k > 0): finite at +∞ but grows without limit as x → −∞: not square-integrable, forbidden.
  3. xe−kx, 0 ≤ x < ∞: single-valued, finite (maximum 1/ke at x = 1/k), continuous, and : allowed; normalised form 2k³ᐟ² xe−kx.SN
  4. e−αx, 0 < x < α: between 1 and e−α², smooth, finite integral: well-behaved. 1/(4 − x), 1 < x < 10: x = 4 lies inside the range, the function goes to +∞ on one side and −∞ on the other: not well-behaved.SN

1.1.3 Normalisation and orthogonality ★★

(1.1.2)
  • Normalisation: multiply ψ by a constant N so that the total probability is 1. A solution of the Schrödinger equation multiplied by any constant is still a solution, so N is always free to choose.SN
  • Orthogonal: two functions are orthogonal when ∫ψi*ψj dτ = 0. Functions that are both normalised and mutually orthogonal are orthonormal (Eq. 1.1.2). Eigenfunctions of a Hermitian operator with different eigenvalues are always orthogonal (Eq. 1.1.5). Worked case: ψ₁ and ψ₂ of a box, in 1.3 Particle in a box.SN

Solved PYQ Geo 2023 Q7(a) 5m: normalise ϕm(ϕ) = A e−imϕ, 0 ≤ ϕ ≤ 2π.

  1. : the probability density is uniform round the ring.
  2. gives A = 1/√(2π); ϕm = (2π)−1/2 e−imϕ.SN
  3. Single-valuedness, ϕ(ϕ + 2π) = ϕ(ϕ), needs e−2πim = 1, so m = 0, ±1, ±2…; different m are orthogonal: ∫ei(m−m′)ϕ dϕ = 0 for m ≠ m′.

1.1.4 Operators and eigenvalue equations ★★★

  • An operator is a rule that turns one function into another: Âf = g. It is linear if Â(c₁f + c₂g) = c₁Âf + c₂Âg (d/dx, x, d²/dx² are linear; √ and “square” are not). All quantum-mechanical operators are linear, so superpositions of solutions are solutions.SN
  • Construction: write the classical expression in x and p, then put x → x× and px → −iħ d/dx. Kinetic energy p²/2m becomes −(ħ²/2m) d²/dx².SN
(1.1.3)
  • If the operator acting on ψ returns the same function multiplied by a constant, ψ is an eigenfunction and the constant is its eigenvalue. The Schrödinger equation Ĥψ = Eψ is the eigenvalue equation of energy.SN
eikx d/dxik eikx ✓ same function × constant: eigenvalue ikcos kxd/dx−k sin kx✗ a different function: not an eigenfunctioncos kxd2 /dx2 −k2  cos kx✓ eigenfunction, eigenvalue −k2Âψ = aψ: the operator returns the SAME function times a number
Fig. 1.1.3 The eigenvalue test: does the operator give back the same function times a number?
Draw it in 60 s: three rows “function → [operator] → result”; tick the rows where the result is the input times a constant.

Solved PYQ Geo 2023 Q1(l) 5m: is ψ = Peikx + Qe−ikx an eigenfunction of d/dx and d²/dx²?

  1. : the sign of the second term has changed, so this is not a constant × ψ: not an eigenfunction of d/dx (unless P = 0 or Q = 0).
  2. : eigenfunction of d²/dx², eigenvalue −k².SN
  3. Meaning: ψ mixes momenta +ħk and −ħk (momentum not sharp) but has one kinetic energy ħ²k²/2m (energy sharp). P = −Q gives sin kx, the particle-in-a-box form.

1.1.5 Hermitian operators ★★★

(1.1.4)
  • Every observable is represented by a Hermitian operator because (i) its eigenvalues are real (a measured value must be real), and (ii) its eigenfunctions with different eigenvalues are orthogonal. x, p̂x, Ĥ are Hermitian; d/dx alone is not.SN

Solved PYQ Geo 2022 Q1(l) 5m: the eigenvalues of a Hermitian operator are real.

  1. Let Âψ = aψ. Multiply by ψ* and integrate: ∫ψ*Âψ dτ = a∫ψ*ψ dτ … (i).
  2. Take the complex conjugate: (Âψ)* = a*ψ*. Use Eq. 1.1.4 with f = g = ψ: ∫ψ*Âψ dτ = ∫ψ(Âψ)* dτ = a*∫ψψ* dτ … (ii).
  3. (i) − (ii): (a − a*)∫|ψ|² dτ = 0. The integral is positive for any non-zero ψ, so a = a*: a is real.SN
  • Orthogonality: ∫ψj*Âψi dτ = ai∫ψj*ψi dτ; by Eq. 1.1.4 the same integral = ∫ψi(Âψj)* dτ = aj∫ψiψj* dτ (aj real). Subtract:SN
(1.1.5)

Solved PYQ Geo 2024 Q8 10m: i d/dx is Hermitian, d/dx is not.

  1. Integrate by parts; the boundary term [f*g] vanishes because well-behaved f, g → 0 at ±∞:
(1.1.6)
  1. The Hermitian test (Eq. 1.1.4) needs +∫g(df/dx)* dx; we got the opposite sign, so d/dx is not Hermitian (it is anti-Hermitian).SN
  2. For i d/dx: left side = i∫f*(dg/dx) dx = −i∫g(df/dx)* dx. Right side = ∫g(i df/dx)* dx = −i∫g(df/dx)* dx since i* = −i. Equal, so i d/dx is Hermitian; so is p̂x = −ħ(i d/dx).SN

Solved PYQ Geo 2025 Q14 20m: Â, B̂ Hermitian, c a constant. (a) c is Hermitian only if c is real; (b)  + B̂ is Hermitian.

  1. (a) Left: ∫f*(cÂg) dτ = c∫f*Âg dτ = c∫g(Âf)* dτ (as  is Hermitian).
  2. Right: ∫g(cÂf)* dτ = c*∫g(Âf)* dτ. The two sides are equal for every f, g only if c = c*, i.e. c real. If c is complex (e.g. c = i: i is anti-Hermitian) they differ, so c is not Hermitian.SN
  3. (b) ∫f*(Â + B̂)g dτ = ∫f*Âg dτ + ∫f*B̂g dτ = ∫g(Âf)* dτ + ∫g(B̂f)* dτ = ∫g[(Â + B̂)f]* dτ: Â + B̂ is Hermitian. So Ĥ = T̂ + V̂ is Hermitian.SN

1.1.6 Commutators and expectation values ★★

(1.1.7)
  • Find a commutator by letting it act on an arbitrary function f(x) and simplifying; then remove f. Two operators commute when the commutator is zero; then they have a common set of eigenfunctions and both observables can be known exactly at the same time Geo 2024 Q1(k) 5m.SN
  • [x̂, d/dx]f = x df/dx − d(xf)/dx = xf′ − (f + xf′) = −f. So [x̂, d/dx] = −1 and [x̂, p̂x] = −iħ(−1) = iħ ≠ 0: position and momentum along the same axis cannot both be sharp. This is the root of ΔxΔpx ≥ ħ/2: see 1.2 Uncertainty principle.SN
fx̂x fd/dxf + x f ′product rulefd/dxf ′x̂x f ′d/dx acts LAST: (d/dx)(x̂ f)x̂ acts LAST: x̂ (d f/dx)x f ′ − (f + x f ′) = − f[x̂, d/dx] = −1
Fig. 1.1.4 The order of the operators matters: the two routes differ by −f.
Draw it in 60 s: two rows of boxes: f → x → d/dx gives f + xf′; f → d/dx → x gives xf′; subtract in a red box: −f.
(1.1.8)
  • Expectation value = the average of a very large number of measurements on identical systems. If ψ is an eigenfunction of Ω̂, every measurement gives the eigenvalue and ⟨Ω⟩ equals it. Always keep the operator between ψ* and ψ.SN
  • For ψ = eikx: ⟨px⟩ = +ħk (an eigenfunction). For ψ = cos kx = ½(eikx + e−ikx): ⟨px⟩ = 0, but each single measurement gives +ħk or −ħk, each with probability ½. For the box ground state ⟨x⟩ = L/2 (solved in 1.3).
  • ★ Beyond the syllabus: for any two observables (Robertson), ΔA = (⟨A²⟩ − ⟨A⟩²)1/2; with [x̂, p̂] = iħ it gives ΔxΔp ≥ ħ/2. Commuting Ĥ, L̂², L̂z are why n, l, ml can be specified together for hydrogen.

1.1.7 How UPSC asks this

Question typeAskedWhat earns the marksTime
Meaning and dimensions of |ψ|²Geo 2024 Q1(l)Born statement; m⁻¹ (1-D), m⁻³ (3-D)4 min
Acceptable / well-behaved functions, test given functionsGeo 2025 Q1(k) Geo 2021 Q1(k) Geo 2023 Q7(b) IFoS 2022 Q1(a)Four conditions, a reason for each; test every function4–7 min
Normalisation; eigenfunction or notGeo 2023 Q7(a) Geo 2023 Q1(l)Integral over the stated range = 1; operate and compare with a constant × ψ4 min
Hermitian proofs: definition, real eigenvalues, i d/dx, cÂ, Â + B̂Geo 2022 Q1(l) Geo 2024 Q8 Geo 2025 Q14Definition first; by parts, boundary term zero; i* = −i4–16 min
CommutatorsGeo 2024 Q1(k)Act on f(x), product rule, drop f4 min

0.8 min per mark (IFoS, Geo-Scientist).

1.1.8 Traps

✗ Calling ψ the probability: the probability density is |ψ|² = ψ*ψ, not ψ².
✗ Dropping the boundary term silently in a Hermitian proof: say it is zero because f, g → 0 at the limits.
✗ “A kink is never allowed”: the slope may jump where V is infinite (box walls), nowhere else.

1.1.9 Where to read

Any one book is enough.

  1. Book 1: P.W. Atkins and J. de Paula, Physical Chemistry, 9th ed., Ch. 7: Born interpretation and acceptable wavefunctions (7.4, p. 262–266), operators and eigenvalues (7.5(b)–(c), p. 267–269), hermitian operators and orthogonality (7.5(d), p. 270–273), expectation values (p. 273–275), commutators (7.6, p. 277–278), postulates (7.7, p. 279).
  2. Book 2: K.L. Kapoor, A Textbook of Physical Chemistry, Vol. IV, 4th ed., Ch. 1, 1.3 “Schrödinger wave theory”: eigenvalue equation (p. 30–31), operators and postulates (p. 32–33), normalisation, orthogonality and Hermitian operators with proofs (p. 34–36), expectation values (p. 36–37).

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