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Eigenvalues and Eigenfunctions in Quantum Mechanics

Eigenvalues represent the permissible measured values of a physical observable obtained by operating with its corresponding quantum mechanical operator on a state wave function. They form the mathematical foundation of quantized energy states and spatial quantization in atomic physics.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Understanding the eigenvalue equation A^psi = a^psi, identification of energy, momentum, and position eigenvalues, and basic operator algebra.

PGT extension

Hermitian operators, real eigenvalues, orthogonal wave functions, expectation values, and application to angular momentum components (L_z, L^2).

Prerequisites

Wave-particle duality and de Broglie hypothesis
Schrödinger wave equation (time-independent and time-dependent)
Linear algebra: matrices and determinants

Definition of Eigenvalue Equation

Mathematically expressed as $A^\psi = a^\psi$, where A^ is a linear differential operator, psi is the eigenfunction, and a is the eigenvalue.
The operator acts on the state function to return the same function multiplied by a constant scalar factor.
The eigenvalues correspond strictly to the experimentally measurable values of the physical observable represented by A^.

Properties of Quantum Mechanical Operators

Observables in quantum mechanics are represented by linear and Hermitian operators.
Hermitian operators guarantee that all calculated eigenvalues are real numbers, which is a physical requirement for any measurable quantity.
Eigenfunctions corresponding to distinct eigenvalues of a Hermitian operator are mutually orthogonal.

Energy Eigenvalues and the Hamiltonian Operator

The Hamiltonian operator H^ represents total energy, yielding the time-independent Schrödinger equation $H^\psi = Epsi$.
For bound systems, the allowed energy eigenvalues E are discrete, leading to quantization.
In atomic systems like the hydrogen atom, energy eigenvalues depend principally on the principal quantum number n.

Angular Momentum Eigenvalues and Spatial Quantization

Orbital angular momentum operators L^2 and L_z possess simultaneous eigenvalues given by l(l+1)hbar^2 and m_lhbar respectively.
The existence of discrete eigenvalues for L_z dictates spatial quantization, restricting the orientation of the angular momentum vector relative to a magnetic field axis.
Total angular momentum combining L-S or j-j coupling similarly follows precise eigenvalue spectra.

Derivations

Eigenvalues of the 1D Infinite Potential Well

Conditions: Particle of mass m confined in a potential V(x) = 0 for 0 < x < L, and V(x) = inf elsewhere.

Write the time-independent Schrödinger equation: -(hbar^2 / 2m) (d^2 psi / d$x^2) = E \psi$
Set up the general solution: psi(x) = A sin(kx) + B cos(kx), where $k^2 = 2mE / hbar^2$
Apply boundary conditions psi(0) = 0 yielding B = 0, and psi(L) = 0 yielding kL = n pi where n = 1, 2, 3...
Substitute k back into the energy expression to extract the discrete eigenvalues.

Result: $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$

Eigenvalues of z-component of Angular Momentum (L_z)

Conditions: In polar coordinates in the xy-plane, operating on the azimuthal wave function.

Write the angular momentum operator in polar coordinates: $L_z = -i \hbar \frac{\partial}{\partial \phi}$
Set up the eigenvalue equation: -i \hbar \frac{d \Phi(\phi)}{d \phi} = m_l \hbar \Phi(\phi)
Separate variables and integrate: \frac{d \Phi}{\Phi} = \frac{i m_l}{\hbar} d \phi
Apply single-valued boundary condition \Phi(\phi + 2\pi) = \Phi(\phi) to restrict m_l to integers.

Result: $L_z = m_l \hbar \text{}$ where } $m_l = 0$, \pm 1, \pm 2, \dots

Formula sheet

Formula / law	Equation	Variables and units	Conditions
General Eigenvalue Equation	$$\hat{A} \psi = a \psi$$	\hat{A} = \text{Operator}, \psi = \text{Eigenfunction}, a = \text{Eigenvalue} Depends on the observable	Valid for linear operators
Time-Independent Schrödinger Equation (Energy Eigenvalue)	$$\hat{H} \psi = E \psi$$	\hat{H} = \text{Hamiltonian operator}, E = \text{Energy eigenvalue} Joules (J) or electron-volts (eV)	Stationary states
Momentum Eigenvalue Equation	$$-i \hbar \frac{\partial}{\partial x} \psi(x) = p \psi(x)$$	\hbar = \text{Reduced Planck constant}, p = \text{Linear momentum} kg m s^-1	Free particle in 1D position representation
Square of Orbital Angular Momentum Eigenvalues	$$\hat{L}^2 Y_{lm} = l(l+1) \hbar^2 Y_{lm}$$	l = \text{Azimuthal quantum number}, $Y_{lm} = \text{Spherical harmonics}$ kg^2 m^4 s^-2	Central potential systems
Z-Component of Angular Momentum Eigenvalues	$$\hat{L}_z Y_{lm} = m_l \hbar Y_{lm}$$	$m_l = \text{Magnetic quantum number}$ kg m^2 s^-1	Spatial quantization condition
Hydrogen Atom Energy Eigenvalues	$$E_n = - \frac{m e^4}{8 \epsilon_0^2 h^2 n^2}$$	m = \text{electron mass}, e = \text{elementary charge}, n = \text{principal quantum number} Joules (J)	Coulombic central potential
Expectation Value of an Operator	$$\langle A \rangle = \frac{\int \psi^* \hat{A} \psi dV}{\int \psi^* \psi dV}$$	\ps$\hat{i}* = \text{Complex conjugate of wave function}$ Units of observable A	Normalized wave function makes denominator 1
Electron Spin Angular Momentum Eigenvalues	$$\hat{S}_z \chi = m_s \hbar \chi$$	$m_s = \text{Spin magnetic quantum number } (\pm 1/2)$ kg m^2 s^-1	Intrinsic spin property

Graphs and experiments

Potential Well Energy Eigenvalues and Wavefunctions

Draw: A 1D box of width L with horizontal energy levels marked E_1, E_2, E_3 and sinusoidal wavefunctions drawn inside the well.

Interpret: Shows discrete energy spacing increasing with n^2 and nodes increasing as n-1.

Exam use: Directly tested in conceptual questions regarding node counts and ground state energy.

Spatial Quantization of Angular Momentum Vector

Draw: A vector cone representation showing permitted orientations of L vector about the z-axis for $l = 1 (m_l = 1$, 0, -1).

Interpret: Demonstrates that L_z can only take specific discrete projections along the magnetic field direction.

Exam use: Used in questions concerning space quantization and Zeeman effect prerequisites.

Hydrogen Energy Level Eigenvalue Convergence

Draw: A vertical ladder diagram of energy levels converging towards E = 0 as n approaches infinity.

Interpret: Illustrates negative bound-state energy eigenvalues and ionization limit at n = inf.

Exam use: Tested in Rydberg transitions and ionization energy calculations.

Solved numericals

TGT core

Calculate the ground state energy eigenvalue (in eV) of an electron confined in a 1D box of length 1 nm.

Identify formula: $E_1 = (1^2 * \pi^2 * hbar^2) / (2 * m * L^2) = (\pi^2 * h^2) / (8 * m * L^2)$
Substitute constants: $h = 6.626 * 10^-34 J s$, $m = 9.11 * 10^-31 kg$, $L = 10^-9 m$
$E_1 = (3.1416^2 * (6.626 * 10^-34)^2) / (8 * 9.11 * 10^-31 * (10^-9)^2)$
Calculate value in Joules and convert to eV by dividing by 1.6 * 10^-19

Answer: 0.377 eV

PGT extension

A quantum system is in an eigenstate of L^2 with l = 2. Find the eigenvalue of L^2 and the possible eigenvalues of L_z.

Use eigenvalue formula for L^2: Lambd$a = l(l+1) hbar^2$
Substitute $l = 2: \lambda = 2(2+1) hbar^2 = 6 hbar^2$
Use eigenvalue formula for L_z: $L_z = m_l hbar$
Determine range of m_l for $l = 2: m_l = -2$, -1, 0, 1, 2
List corresponding L_z eigenvalues: -2hbar, -hbar, 0, hbar, 2hbar

Answer: L^2 eigenvalu$e = 6\hbar^2; L_z eigenvalues = -2\hbar$, -\hbar, 0, \hbar, 2\hbar

PGT extension

Verify if the function psi(x) = e^(2i x) is an eigenfunction of the momentum operator $p_x = -i hbar (d/dx)$, and find its momentum eigenvalue.

Apply momentum operator to the function: p_x ps$i = -i \hbar \frac{d}{dx} (e^{2i x})$
Differentiate: \frac{d}{dx} (e^{2i x}) = 2i e^{2i x}
Multiply by -i \hbar: p_x ps$i = -i \hbar (2i e^{2i x})$
Simplify using $i^2 = -1: p_x \psi = (-i)(2i) \hbar e^{2i x} = 2 \hbar e^{2i x}$
Compare with eigenvalue form $A^\psi = a \psi$ to identify a

Answer: Yes, it is an eigenfunction with momentum eigenvalue p = 2\hbar

Assumptions and limitations

Operators representing physical observables must be linear and Hermitian to ensure real eigenvalues, which limits the mathematical form of admissible operators.

Common misconceptions

Common error: Any arbitrary wave function is always an eigenfunction of any quantum operator.

Correct: A wave function is only an eigenfunction for a specific operator if its application returns the exact same function multiplied by a constant.

Superposition states do not return a single eigenvalue upon measurement, but rather yield probabilistic distributions of multiple eigenvalues.

Common error: Eigenvalues in quantum mechanics can take any continuous real value for bound microscopic systems.

Correct: Bound systems possess discrete, quantized sets of eigenvalues.

Boundary conditions force quantization, restricting allowable eigenvalues to discrete spectra.

Quick revision

The eigenvalue equation is represented as $A^\psi = apsi$, where a denotes the measurable value.
General Eigenvalue Equation: \hat{A} \psi = a \psi
Time-Independent Schrödinger Equation (Energy Eigenvalue): \hat{H} \psi = E \psi

Must remember

The general eigenvalue equation is given by $\hat{A} \psi = a \psi$, where $\hat{A}$ is a linear differential operator, $\psi$ is the eigenfunction, and $a$ is the eigenvalue representing a measurable physical quantity.
Quantum mechanical observables are represented by linear and Hermitian operators to guarantee that all calculated eigenvalues are real numbers.
Eigenfunctions corresponding to distinct eigenvalues of a Hermitian operator are mutually orthogonal.
The time-independent Schrödinger equation $\hat{H} \psi = E \psi$ yields energy eigenvalues, which are discrete for bound systems due to boundary conditions.
Orbital angular momentum operators $\hat{L}^2$ and $\hat{L}_z$ possess simultaneous eigenvalues given by $l(l+1)\hbar^2$ and $m_l\hbar$ respectively, leading to spatial quantization.

Glossary

Eigenvalue EquationAn equation of the form $\hat{A} \psi = a \psi$ where an operator acting on a function returns a constant multiple of the same function.

Hermitian OperatorAn operator that satisfies the condition ensuring all its eigenvalues are strictly real numbers, representing physical observables.

Hamiltonian Operator ($\hat{H}$)The operator corresponding to the total energy of the quantum system, appearing in the Schrödinger equation.

Spatial QuantizationThe restriction of the orientation of angular momentum vectors relative to a reference axis, dictated by discrete eigenvalues of $\hat{L}_z$.

Expectation ValueThe statistical average of outcomes of multiple measurements of an observable, calculated via $\langle A \rangle = \frac{\int \psi^* \hat{A} \psi dV}{\int \psi^* \psi dV}$.

Comparison of Angular Momentum Operators $\hat{L}^2$ and $\hat{L}_z$

Feature	$\hat{L}^2$ (Total Angular Momentum Squared)	$\hat{L}_z$ (Z-Component of Angular Momentum)
Eigenvalue Expression	$l(l+1)\hbar^2$	$m_l \hbar$
Quantum Numbers Involved	Azimuthal quantum number $l$	Magnetic quantum number $m_l$
Physical Significance	Represents the magnitude squared of the orbital angular momentum	Represents spatial orientation relative to the z-axis (spatial quantization)

Practice Quiz

Choose an option to reveal the explanation.

1. What is the mathematical form of the general eigenvalue equation in quantum mechanics?

Explanation: The eigenvalue equation is expressed as $\hat{A} \psi = a \psi$, where $\hat{A}$ is the operator, $\psi$ is the eigenfunction, and $a$ is the scalar eigenvalue.

2. Why must quantum mechanical operators representing physical observables be Hermitian?

Explanation: Hermitian operators guarantee that all calculated eigenvalues are real numbers, which is a fundamental physical requirement for any measurable quantity.

3. What are the eigenvalues of the orbital angular momentum operator $\hat{L}^2$?

Explanation: The square of orbital angular momentum operators $\hat{L}^2$ possesses eigenvalues given by $l(l+1)\hbar^2$.

4. What does the Hamiltonian operator $\hat{H}$ represent in quantum mechanics?

Explanation: The Hamiltonian operator $\hat{H}$ represents the total energy of the system, yielding the time-independent Schrödinger equation $\hat{H} \psi = E \psi$.

5. What physical phenomenon is dictated by the discrete eigenvalues of the $\hat{L}_z$ operator?

Explanation: The existence of discrete eigenvalues for $\hat{L}_z$ (namely $m_l \hbar$) dictates spatial quantization, restricting the orientation of the angular momentum vector relative to a magnetic field axis.

6. What is the momentum eigenvalue equation for a free particle in 1D position representation?

Explanation: The linear momentum operator in the position representation is $-i\hbar \frac{\partial}{\partial x}$, giving the eigenvalue equation $-i \hbar \frac{\partial}{\partial x} \psi(x) = p \psi(x)$.

7. What is the relationship between eigenfunctions corresponding to distinct eigenvalues of a Hermitian operator?

Explanation: Eigenfunctions corresponding to distinct eigenvalues of a Hermitian operator are mutually orthogonal.

8. In the hydrogen atom energy eigenvalue formula $E_n = - \frac{m e^4}{8 \epsilon_0^2 h^2 n^2}$, what does $n$ represent?

Explanation: In atomic systems like the hydrogen atom, energy eigenvalues depend principally on the principal quantum number $n$.

9. What is the correct expression for the expectation value of an operator $\hat{A}$ for a normalized wave function?

Explanation: For a normalized wave function where the denominator $\int \psi^* \psi dV = 1$, the expectation value is given by $\langle A \rangle = \int \psi^* \hat{A} \psi dV$.

10. Which quantum number specifies the eigenvalues of the electron spin z-component operator $\hat{S}_z$?

Explanation: The spin magnetic quantum number $m_s$ (taking values $\pm 1/2$) specifies the eigenvalues of $\hat{S}_z \chi = m_s \hbar \chi$.

11. Why do bound systems possess discrete energy eigenvalues rather than continuous values?

Explanation: Boundary conditions force quantization, restricting allowable eigenvalues to discrete spectra for bound systems.

12. If an arbitrary wave function is expressed as a superposition of different eigenfunctions of an operator, what happens when the operator is applied?

Explanation: Superposition states do not return a single eigenvalue upon measurement, but rather yield probabilistic distributions of multiple eigenvalues.

13. What are the units of energy eigenvalues in quantum mechanics?

Explanation: Energy eigenvalues represent energy levels, with standard units of Joules (J) or electron-volts (eV).

14. What is the eigenvalue equation for the z-component of orbital angular momentum?

Explanation: The z-component of angular momentum has eigenvalues given by $m_l \hbar$, expressed as $\hat{L}_z Y_{lm} = m_l \hbar Y_{lm}$.

15. Which of the following statements is correct regarding an operator equation $\hat{A} \psi = a \psi$?

Explanation: A wave function is only an eigenfunction for a specific operator if its application returns the exact same function multiplied by a constant scalar factor $a$.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The eigenvalues of the Hermitian operator representing a physical observable must be:

Explanation: Hermitian operators guarantee that all calculated eigenvalues are real numbers, which corresponds to experimentally measurable physical quantities.

2. What is the eigenvalue of the z-component of orbital angular momentum operator $\hat{L}_z$ when operating on spherical harmonics $Y_{lm}$?

Explanation: The eigenvalue equation for $\hat{L}_z$ is $\hat{L}_z Y_{lm} = m_l \hbar Y_{lm}$, where $m_l \hbar$ is the eigenvalue.

3. The time-independent Schrödinger equation $\hat{H} \psi = E \psi$ is an example of:

Explanation: The time-independent Schrödinger equation is the fundamental eigenvalue equation for the Hamiltonian operator $\hat{H}$, where $E$ represents the energy eigenvalue.

4. For a bound quantum mechanical system, the allowed energy eigenvalues are:

Explanation: For bound systems, boundary conditions force the allowed energy eigenvalues $E$ to be discrete, leading to quantization.

5. If $\psi_1$ and $\psi_2$ are two eigenfunctions of a Hermitian operator corresponding to distinct eigenvalues $a_1$ and $a_2$, then:

Explanation: A key theorem of Hermitian operators states that eigenfunctions corresponding to distinct eigenvalues are mutually orthogonal.

6. What does the operator expression $\hat{L}^2 Y_{lm} = l(l+1) \hbar^2 Y_{lm}$ signify?

Explanation: The equation represents the square of orbital angular momentum eigenvalues given by $l(l+1)\hbar^2$ for central potential systems.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which of the following conditions is necessary for a wave function $\psi$ to be an eigenfunction of an operator $\hat{A}$?

Explanation: By definition, $\psi$ is an eigenfunction of $\hat{A}$ if applying $\hat{A}$ returns the exact same function multiplied by a constant scalar factor.

2. What is the physical significance of eigenvalues in quantum mechanics?

Explanation: The eigenvalues correspond strictly to the experimentally measurable values of the physical observable represented by the operator.

3. What are the eigenvalues for the square of orbital angular momentum $\hat{L}^2$ in terms of azimuthal quantum number $l$?

Explanation: The orbital angular momentum operator $\hat{L}^2$ possesses eigenvalues given by $l(l+1)\hbar^2$.

4. What does the expectation value formula $\langle A \rangle = \frac{\int \psi^* \hat{A} \psi dV}{\int \psi^* \psi dV}$ simplify to when the wave function is normalized?

Explanation: When the wave function is normalized, the denominator $\int \psi^* \psi dV$ equals 1, simplifying the expression to $\int \psi^* \hat{A} \psi dV$.

5. In the hydrogen atom energy eigenvalue formula $E_n = - \frac{m e^4}{8 \epsilon_0^2 h^2 n^2}$, how does energy scale with the principal quantum number $n$?

Explanation: Energy eigenvalues for the hydrogen atom depend inversely on the square of the principal quantum number ($n^2$).

6. Which operator equation represents the linear momentum eigenvalue equation for a 1D free particle?

Explanation: The momentum operator in position representation is $-i\hbar \frac{\partial}{\partial x}$, yielding $-i \hbar \frac{\partial}{\partial x} \psi(x) = p \psi(x)$.

7. What is the nature of eigenvalues for electron spin z-component operator $\hat{S}_z$?

Explanation: Electron spin eigenvalues for $\hat{S}_z$ are given by $m_s \hbar$ where $m_s = \pm 1/2$, resulting in $\pm \frac{1}{2} \hbar$.

8. Why do superposition states fail to yield a single sharp eigenvalue upon measurement?

Explanation: Superposition states represent a mixture of multiple eigenfunctions, leading to probabilistic outcomes rather than a single definite eigenvalue.

9. What property of Hermitian operators ensures that expectation values of observables are always real?

Explanation: Since Hermitian operators possess real eigenvalues, the expectation values calculated from them are also guaranteed to be real numbers.

10. What physical restriction leads directly to spatial quantization in quantum mechanics?

Explanation: Single-valuedness and periodic boundary conditions for angular wave functions restrict $\hat{L}_z$ eigenvalues to discrete values $m_l \hbar$, causing spatial quantization.