Unit 3: Schrödinger’s Equation and Operators

1. Time-Dependent & Time-Independent Schrödinger Equations

1.1 Time-Dependent Schrödinger Equation (TDSE)

The Time-Dependent Schrödinger Equation (TDSE) is the fundamental equation of motion in non-relativistic quantum mechanics. It describes how the quantum state, represented by the wavefunction Ψ(r, t), evolves continuously over time.

iℏ (∂Ψ(r, t) / ∂t) = Ĥ Ψ(r, t)

For a non-relativistic particle of mass m moving in a potential field V(r, t), the one-dimensional and three-dimensional time-dependent Schrödinger equations are expressed as:

One-Dimensional (1D) TDSE:
iℏ (∂Ψ(x, t) / ∂t) = - (ℏ² / 2m) (∂²Ψ(x, t) / ∂x²) + V(x, t) Ψ(x, t)

Three-Dimensional (3D) TDSE:
iℏ (∂Ψ(r, t) / ∂t) = - (ℏ² / 2m) ∇²Ψ(r, t) + V(r, t) Ψ(r, t)

Where ℏ = h / 2π is the reduced Planck constant, i is the imaginary unit (i² = -1), and ∇² is the Laplacian operator defined in Cartesian coordinates as ∇² = (∂² / ∂x²) + (∂² / ∂y²) + (∂² / ∂z²).

1.2 Non-Relativistic Nature of the Schrödinger Equation

The equation is non-relativistic because it is constructed using the classical kinetic energy relation T = p² / 2m rather than Einstein's relativistic energy relation E² = p²c² + m²c⁴. Consequently, space coordinates enter as second-order derivatives, while time enters as a first-order derivative.

1.3 Derivation of Time-Independent Schrödinger Equation (TISE)

When the potential energy V(r) depends only on spatial coordinates and is independent of time t, we can solve the TDSE using the method of separation of variables.

Step 1: Assume the wavefunction is factorable into a spatial function ψ(r) and a temporal function ϕ(t):

Ψ(r, t) = ψ(r) ϕ(t)

Step 2: Substitute this separable wavefunction into the 3D TDSE:

iℏ ψ(r) (dϕ(t) / dt) = - (ℏ² / 2m) ϕ(t) ∇²ψ(r) + V(r) ψ(r) ϕ(t)

Step 3: Divide the entire equation by Ψ(r, t) = ψ(r) ϕ(t):

iℏ (1 / ϕ(t)) (dϕ(t) / dt) = - (ℏ² / 2m) (1 / ψ(r)) ∇²ψ(r) + V(r)

The left side depends solely on time t, while the right side depends solely on position r. For these two sides to be equal for all r and t, both sides must equal a separation constant, defined as the total energy E.

Step 4: Solve the temporal part:

iℏ (1 / ϕ(t)) (dϕ(t) / dt) = E → (dϕ(t) / dt) = - (iE / ℏ) ϕ(t)

Integrating gives the temporal solution:

ϕ(t) = e^{-iEt / ℏ}

Step 5: Equate the spatial part to E:

- (ℏ² / 2m) (1 / ψ(r)) ∇²ψ(r) + V(r) = E

Multiplying by ψ(r) yields the Time-Independent Schrödinger Equation (TISE):

- (ℏ² / 2m) ∇²ψ(r) + V(r) ψ(r) = E ψ(r)
or
Ĥ ψ(r) = E ψ(r)

1.4 Stationary States

States represented by Ψ(r, t) = ψ(r) e^{-iEt / ℏ} are called stationary states because the probability density |Ψ(r, t)|² remains constant over time:

|Ψ(r, t)|² = Ψ^*(r, t) Ψ(r, t) = [ψ^*(r) e^{iEt / ℏ}] [ψ(r) e^{-iEt / ℏ}] = |ψ(r)|²

2. Expectation Value of Dynamical Variables

2.1 Concept and Definition

In quantum mechanics, a single measurement of an observable A yields one of its eigenvalues. The expectation value ⟨A⟩ is the statistical ensemble average of measurement outcomes obtained from an infinite number of identically prepared systems in state Ψ.

⟨A⟩ = ∫_-∞^∞ Ψ^* Â Ψ dτ / ∫_-∞^∞ Ψ^* Ψ dτ

If the wavefunction Ψ is normalized (∫_-∞^∞ Ψ^* Ψ dτ = 1), the expression simplifies to:

⟨A⟩ = ∫_-∞^∞ Ψ^* Â Ψ dτ

2.2 Expectation Values of Specific Observables

• Position: ⟨x⟩ = ∫_-∞^∞ Ψ^*(x, t) x Ψ(x, t) dx
• Momentum: ⟨p_x⟩ = ∫_-∞^∞ Ψ^*(x, t) (-iℏ ∂ / ∂x) Ψ(x, t) dx
• Kinetic Energy: ⟨T⟩ = ∫_-∞^∞ Ψ^*(x, t) (- (ℏ² / 2m) ∂² / ∂x²) Ψ(x, t) dx
• Total Energy: ⟨E⟩ = ∫_-∞^∞ Ψ^*(x, t) (iℏ ∂ / ∂t) Ψ(x, t) dx

2.3 Exam Notes & Common Mistakes

Important Note: The operator  MUST be sandwiched between Ψ^* and Ψ because differential operators act specifically on the wavefunction placed to their right.

Common Mistake: Writing ⟨A⟩ = ∫  |Ψ|² dτ is incorrect when  is a differential operator, such as momentum or kinetic energy, because differential operators do not commute with simple multiplication.

3. Quantum Mechanical Operators and Operator Algebra

3.1 Definition of Quantum Operators

An operator is a mathematical instruction that transforms one function into another function: Â ψ(x) = ϕ(x). Quantum mechanics postulates that every physically measurable classical observable corresponds to a linear Hermitian operator.

3.2 Fundamental Rules of Operator Algebra

• Addition: (Â + B̂) ψ = Â ψ + B̂ ψ
• Commutative Addition: Â + B̂ = B̂ + Â
• Multiplication: ( B̂) ψ =  (B̂ ψ) — Operator B̂ acts first on ψ, and then operator  acts on the resulting function.
• Associative Law: Â (B̂ Ĉ) = (Â B̂) Ĉ
• Distributive Law: Â (B̂ + Ĉ) = Â B̂ + Â Ĉ
• Identity Operator (Î): Î ψ = ψ
• Null Operator (0̂): 0̂ ψ = 0
• Non-Commutativity in General: In general, Â B̂ ≠ B̂ Â. Order of application is critical.

4. Linear Operators, Eigenvalues, and Eigenfunctions

4.1 Linear Operators

An operator  is defined as a linear operator if it satisfies two essential conditions for any complex constants c₁ and c₂ and any wavefunctions ψ₁ and ψ₂:

 (c₁ ψ₁ + c₂ ψ₂) = c₁  ψ₁ + c₂  ψ₂

All quantum mechanical operators representing physical observables are linear.

4.2 Eigenvalue Equation, Eigenfunctions, and Eigenvalues

When an operator  acts on a specific non-zero function ψ, returning the exact same function multiplied by a scalar constant λ, this mathematical relation is called an eigenvalue equation:

 ψ = λ ψ

• Eigenfunction (ψ): The non-trivial wavefunction that retains its spatial form under the operation of Â.
• Eigenvalue (λ): The scalar multiplier corresponding to the measured magnitude of observable A.

4.3 Degeneracy

If two or more linearly independent eigenfunctions ψ₁, ψ₂, ..., ψ_k share the exact same eigenvalue λ for an operator Â, the eigenvalue λ is said to be k-fold degenerate. If only one eigenfunction corresponds to λ, it is non-degenerate.

4.4 Orthonormality Condition

Eigenfunctions of quantum operators are usually normalized and mutually orthogonal. Combined, this is expressed using the Kronecker delta δ_ij:

∫_-∞^∞ ψ_i^* ψ_j dτ = δ_ij = 1 if i = j, and 0 if i ≠ j

5. Specific Quantum Mechanical Operators

6. Commutation Relations

6.1 Definition of Commutator

The commutator of two operators  and B̂ is defined as:

[Â, B̂] = Â B̂ - B̂ Â

If [Â, B̂] = 0, the operators are said to commute. If [Â, B̂] ≠ 0, they do not commute.

6.2 Physical Significance of Commutators

• Commuting Observables: If two operators commute ([Â, B̂] = 0), their physical observables can be measured simultaneously with arbitrary precision, and they share a complete set of simultaneous eigenfunctions.
• Non-Commuting Observables: If [Â, B̂] ≠ 0, the observables cannot be measured simultaneously with absolute precision, leading directly to the generalized Heisenberg Uncertainty Principle: ΔA ΔB ≥ ½ |⟨[Â, B̂]⟩|.

6.3 Fundamental Algebra Properties of Commutators

• Anti-symmetry: [Â, B̂] = -[B̂, Â]
• Linearity: [Â, B̂ + Ĉ] = [Â, B̂] + [Â, Ĉ]
• Distributive Product Rule 1: [Â, B̂ Ĉ] = [Â, B̂] Ĉ + B̂ [Â, Ĉ]
• Distributive Product Rule 2: [Â B̂, Ĉ] = Â [B̂, Ĉ] + [Â, Ĉ] B̂
• Jacobi Identity: [Â, [B̂, Ĉ]] + [B̂, [Ĉ, Â]] + [Ĉ, [Â, B̂]] = 0

6.4 Canonical Commutation Relation: Step-by-Step Derivation of [x, p̂_x]

To evaluate [x, p̂_x], let it act on an arbitrary test function ψ(x):

[x, p̂_x] ψ = (x p̂_x - p̂_x x) ψ
= x (-iℏ (∂ψ / ∂x)) - (-iℏ (∂(x ψ) / ∂x))

Applying the product rule of differentiation to ∂(x ψ) / ∂x:

∂(x ψ) / ∂x = ψ + x (∂ψ / ∂x)

Substituting back:

[x, p̂_x] ψ = -iℏ x (∂ψ / ∂x) + iℏ [ψ + x (∂ψ / ∂x)]
= -iℏ x (∂ψ / ∂x) + iℏ ψ + iℏ x (∂ψ / ∂x)
= iℏ ψ

Removing the test wavefunction ψ yields the canonical relation:

[x, p̂_x] = iℏ

Similarly, for distinct dimensions:

[x, p̂_y] = 0, [y, p̂_z] = 0, [x, y] = 0, [p̂_x, p̂_y] = 0

6.5 Angular Momentum Commutation Relations

Using components L̂_x, L̂_y, L̂_z:

[L̂_x, L̂_y] = iℏ L̂_z
[L̂_y, L̂_z] = iℏ L̂_x
[L̂_z, L̂_x] = iℏ L̂_y

For total angular momentum squared L̂²:

[L̂², L̂_x] = 0, [L̂², L̂_y] = 0, [L̂², L̂_z] = 0

Observation: L̂² commutes with all individual angular momentum components, but individual components do not commute among themselves. Hence, one can measure L̂² and L̂_z simultaneously, but not L̂_x and L̂_y concurrently.

7. Hermitian Operators and Their Properties

7.1 Definition of Hermitian Operators

An operator  is defined as a Hermitian operator if it satisfies the integral condition for all well-behaved wavefunctions ψ_i and ψ_j:

∫_-∞^∞ ψ_i^* (Â ψ_j) dτ = ∫_-∞^∞ (Â ψ_i)^* ψ_j dτ

In Dirac bra-ket notation, this is expressed as ⟨ψ_i|Â|ψ_j⟩ = ⟨Â ψ_i|ψ_j⟩.

7.2 Proof: Momentum Operator p̂_x = -iℏ (∂ / ∂x) is Hermitian

To prove p̂_x is Hermitian, consider the integral:

∫_-∞^∞ ψ_i^* (-iℏ (∂ψ_j / ∂x)) dx = -iℏ ∫_-∞^∞ ψ_i^* (∂ψ_j / ∂x) dx

Integrate by parts using u = ψ_i^* and dv = (∂ψ_j / ∂x) dx:

= -iℏ [ψ_i^* ψ_j]_-∞^∞ + iℏ ∫_-∞^∞ (∂ψ_i^* / ∂x) ψ_j dx

Because physical wavefunctions vanish at infinity (ψ → 0 as x → ±∞), the boundary term vanishes:

= ∫_-∞^∞ (iℏ (∂ψ_i^* / ∂x)) ψ_j dx = ∫_-∞^∞ (-iℏ (∂ψ_i / ∂x))^* ψ_j dx = ∫_-∞^∞ (p̂_x ψ_i)^* ψ_j dx

Thus, p̂_x is strictly Hermitian.

7.3 Fundamental Theorem 1: Eigenvalues of Hermitian Operators are Real

Theorem Statement: All eigenvalues of a Hermitian operator are real numbers.

Proof: Let  be a Hermitian operator with eigenvalue equation  ψ = λ ψ.

Consider the inner product integral:

∫_-∞^∞ ψ^* Â ψ dτ = ∫_-∞^∞ ψ^* (λ ψ) dτ = λ ∫_-∞^∞ ψ^* ψ dτ

Now consider the right side of the Hermitian definition:

∫_-∞^∞ (Â ψ)^* ψ dτ = ∫_-∞^∞ (λ ψ)^* ψ dτ = λ^* ∫_-∞^∞ ψ^* ψ dτ

Since  is Hermitian, these two integrals are equal:

λ ∫_-∞^∞ ψ^* ψ dτ = λ^* ∫_-∞^∞ ψ^* ψ dτ → (λ - λ^*) ∫_-∞^∞ |ψ|² dτ = 0

Since ψ is a valid non-zero wavefunction, ∫_-∞^∞ |ψ|² dτ ≠ 0. Therefore:

λ - λ^* = 0 → λ = λ^*

This proves that the eigenvalue λ is strictly a real number.

7.4 Fundamental Theorem 2: Orthogonality of Eigenfunctions

Theorem Statement: Eigenfunctions corresponding to distinct (non-degenerate) eigenvalues of a Hermitian operator are mutually orthogonal.

Proof: Let  ψ₁ = λ₁ ψ₁ and  ψ₂ = λ₂ ψ₂, where λ₁ ≠ λ₂.

Evaluate the integral ∫_-∞^∞ ψ₁^* Â ψ₂ dτ:

∫_-∞^∞ ψ₁^* Â ψ₂ dτ = λ₂ ∫_-∞^∞ ψ₁^* ψ₂ dτ

Now evaluate the Hermitian conjugate expression ∫_-∞^∞ (Â ψ₁)^* ψ₂ dτ:

∫_-∞^∞ (Â ψ₁)^* ψ₂ dτ = ∫_-∞^∞ (λ₁ ψ₁)^* ψ₂ dτ = λ₁^* ∫_-∞^∞ ψ₁^* ψ₂ dτ

Since λ₁ is real (λ₁^* = λ₁), and  is Hermitian, set the two expressions equal:

λ₂ ∫_-∞^∞ ψ₁^* ψ₂ dτ = λ₁ ∫_-∞^∞ ψ₁^* ψ₂ dτ → (λ₁ - λ₂) ∫_-∞^∞ ψ₁^* ψ₂ dτ = 0

Since λ₁ ≠ λ₂, (λ₁ - λ₂) ≠ 0. Consequently:

∫_-∞^∞ ψ₁^* ψ₂ dτ = 0

This proves that the eigenfunctions ψ₁ and ψ₂ are orthogonal.