Schrödinger Equation (Time-Independent): The Cornerstone of Quantum Bound States

Table of Contents

1. Introduction
2. The Schrödinger Equation Framework
3. Time-Dependent vs Time-Independent Equations
4. Derivation from Separation of Variables
5. Mathematical Form of the Time-Independent Schrödinger Equation
6. Physical Interpretation
7. Boundary Conditions and Normalization
8. One-Dimensional Potential Wells
9. Particle in a Box
10. Quantum Harmonic Oscillator
11. Finite Square Well
12. Tunneling and Potential Barriers
13. Energy Quantization
14. Eigenstates and Orthogonality
15. Operators and Commutators in TISE
16. Higher-Dimensional Systems
17. Importance in Atomic and Molecular Physics
18. Conclusion

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1. Introduction

The time-independent Schrödinger equation (TISE) lies at the heart of non-relativistic quantum mechanics. It provides a framework for understanding quantum systems in stationary states, such as electrons in atoms and molecules. Solving the TISE yields quantized energy levels and wavefunctions, which form the foundation of our understanding of quantum structure.

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2. The Schrödinger Equation Framework

Proposed by Erwin Schrödinger in 1926, the equation governs the wavefunction \( \psi \), a complex-valued function whose modulus squared represents the probability density of finding a particle.

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3. Time-Dependent vs Time-Independent Equations

Time-Dependent Schrödinger Equation (TDSE):

\[
i\hbar \frac{\partial}{\partial t} \Psi(x, t) = \hat{H} \Psi(x, t)
\]

Applicable to all quantum systems. Solutions describe full time evolution.

Time-Independent Schrödinger Equation (TISE):

\[
\hat{H} \psi(x) = E \psi(x)
\]

Applies when the potential \( V(x) \) is not explicitly time-dependent. Solutions give stationary states with definite energy.

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4. Derivation from Separation of Variables

Assume:

\[
\Psi(x, t) = \psi(x) e^{-iEt/\hbar}
\]

Substitute into TDSE and divide by \( \Psi \):

\[
\hat{H} \psi(x) = E \psi(x)
\]

This gives the time-independent equation.

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5. Mathematical Form of the Time-Independent Schrödinger Equation

In one dimension:

\[
\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)
\]

Where:

• \( \hbar \) is the reduced Planck constant
• \( m \) is the mass of the particle
• \( V(x) \) is the potential energy
• \( E \) is the total energy of the system

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6. Physical Interpretation

• \( \psi(x) \) is a probability amplitude
• \( |\psi(x)|^2 \) is the probability density
• The equation represents a balance between kinetic and potential energy

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7. Boundary Conditions and Normalization

Wavefunctions must satisfy:

• Continuity: \( \psi(x) \) and \( \frac{d\psi}{dx} \) must be continuous
• Normalization:
  \[
  \int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1
  \]
• Boundary conditions: depend on potential and geometry

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8. One-Dimensional Potential Wells

Potential wells are regions where \( V(x) < E \).

• Infinite potential well: simplest example
• Finite well: allows tunneling
• Harmonic oscillator: parabolic potential

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9. Particle in a Box

Infinite square well with \( V(x) = 0 \) for \( 0 < x < L \), \( V = \infty \) elsewhere:

\[
\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right)
\]

\[
E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}
\]

Quantized energies due to boundary conditions.

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10. Quantum Harmonic Oscillator

Potential:

\[
V(x) = \frac{1}{2} m \omega^2 x^2
\]

Solutions:

\[
E_n = \hbar \omega \left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \dots
\]

Wavefunctions involve Hermite polynomials.

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11. Finite Square Well

Potential:

\[
V(x) = \begin{cases}
-V_0, & |x| < a \ 0, & |x| > a
\end{cases}
\]

• Bound states for \( E < 0 \)
• Discrete number of eigenvalues
• Tunneling into classically forbidden region

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12. Tunneling and Potential Barriers

For a barrier \( V(x) > E \), classical physics predicts no penetration. Quantum mechanics allows tunneling:

• \( \psi(x) \) decays exponentially in barrier
• Transmission probability nonzero
• Crucial in nuclear decay, electronics (tunnel diodes)

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13. Energy Quantization

Solving the TISE results in discrete energy eigenvalues:

• Arises due to boundary conditions and continuity
• Explains spectral lines and stability of matter

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14. Eigenstates and Orthogonality

• Solutions \( \psi_n(x) \) are eigenfunctions
• Correspond to eigenvalues \( E_n \)
• Orthogonality condition:

\[
\int \psi_m^*(x) \psi_n(x) dx = \delta_{mn}
\]

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15. Operators and Commutators in TISE

• Hamiltonian \( \hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}) \)
• Position \( \hat{x} \): multiplication
• Momentum \( \hat{p} = -i\hbar \frac{d}{dx} \)

Key commutator:

\[
[\hat{x}, \hat{p}] = i\hbar
\]

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16. Higher-Dimensional Systems

In 3D:

\[
-\frac{\hbar^2}{2m} \nabla^2 \psi(\vec{r}) + V(\vec{r}) \psi(\vec{r}) = E \psi(\vec{r})
\]

Used in:

• Atoms (hydrogen)
• Molecules
• Solid-state systems

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17. Importance in Atomic and Molecular Physics

TISE describes:

• Atomic orbitals
• Molecular vibrations and rotations
• Chemical bonding
• Energy quantization in nanoparticles and quantum dots

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18. Conclusion

The time-independent Schrödinger equation is a cornerstone of quantum mechanics. It captures the essence of stationary quantum systems and introduces quantized energy levels, tunneling, and wavefunction structure. Mastery of the TISE is critical for any deeper study of quantum physics, chemistry, and nanotechnology.

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