What is Schrödinger equation

⚛️ The Schrödinger Equation: A Quantum Foundation in Physical Chemistry

One of the most fundamental concepts in physical chemistry is the Schrödinger equation, which lies at the heart of quantum mechanics. It provides a mathematical framework to describe the behavior of particles—such as electrons—at the atomic and subatomic level. Understanding this equation is key to explaining chemical bonding, molecular structure, and spectroscopy.

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📐 The Schrödinger Equation: Time-Independent Form

The time-independent Schrödinger equation for a single particle in one dimension is written as: H^ψ=Eψ\hat{H}\psi = E\psiH^ψ=Eψ

Where:

• H^\hat{H}H^ is the Hamiltonian operator (total energy operator),
• ψ\psiψ is the wavefunction,
• EEE is the energy of the system.

The Hamiltonian typically consists of two parts: H^=T^+V^\hat{H} = \hat{T} + \hat{V}H^=T^+V^

where T^\hat{T}T^ is the kinetic energy operator and V^\hat{V}V^ is the potential energy operator.

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🌊 What is a Wavefunction?

The wavefunction ψ(x)\psi(x)ψ(x) contains all the information about a quantum system. While ψ\psiψ itself is not directly observable, its square ∣ψ(x)∣2|\psi(x)|^2∣ψ(x)∣2 gives the probability density of finding a particle at a position xxx. This probabilistic interpretation marks a shift from classical mechanics to quantum mechanics.

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📦 Particle in a Box: A Classic Example

One of the simplest and most illustrative problems is the particle-in-a-box (also known as the infinite potential well). In this model:

• The potential energy V(x)V(x)V(x) is zero inside the box (0 < x < L),
• And infinite outside the box.

The allowed wavefunctions are: ψn(x)=2Lsin⁡(nπxL)\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)ψn​(x)=L2​​sin(Lnπx​)

The corresponding energy levels are quantized: En=n2h28mL2E_n = \frac{n^2 h^2}{8mL^2}En​=8mL2n2h2​

where n=1,2,3,…n = 1, 2, 3, \ldotsn=1,2,3,… is the quantum number, hhh is Planck’s constant, mmm is the mass of the particle, and LLL is the box length.

This model illustrates that energy is not continuous, but quantized.

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🧪 Application in Physical Chemistry

In physical chemistry, the Schrödinger equation is used to:

• Predict molecular orbitals (in atomic and molecular systems),
• Understand vibrational and rotational energy levels in spectroscopy,
• Model electron density in molecules (e.g., in ab initio or DFT calculations).

Although exact solutions exist only for simple systems, approximate methods (like Hartree-Fock) rely on the foundation laid by Schrödinger’s work.

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🧠 Quiz: Schrödinger Equation Basics

1. What does the wavefunction ψ(x)\psi(x)ψ(x) represent in quantum mechanics?
   A. Kinetic energy of a particle
   B. Exact location of a particle
   C. Probability density function
   D. Amplitude of the particle’s motion
   Answer: C
2. Which of the following is true about the particle-in-a-box model?
   A. Energy levels are continuous
   B. The wavefunction is zero at the walls
   C. Potential energy is constant everywhere
   D. Only even quantum numbers are allowed
   Answer: B
3. In the Schrödinger equation, what does the Hamiltonian operator represent?
   A. Momentum of the system
   B. Probability of a state
   C. Total energy operator
   D. Position of the particle
   Answer: C
4. Which variable(s) affect the energy levels in the particle-in-a-box model?
   A. Mass of the particle
   B. Length of the box
   C. Quantum number nnn
   D. All of the above
   Answer: D
5. Why are the energy levels in a quantum system quantized?
   A. Due to thermal motion
   B. Because wavefunctions must satisfy boundary conditions
   C. Because the system is at low temperature
   D. Due to uncertainty principle
   Answer: B