1.3 Schrödinger equation and quantum states

The Schrödinger equation is the cornerstone of quantum mechanics, describing how quantum states evolve over time. It comes in two forms: time-dependent and time-independent, each crucial for understanding different aspects of quantum systems and their behavior.

Solving the Schrödinger equation reveals fundamental quantum phenomena like quantized energy levels and wavefunctions. These solutions provide insights into real-world systems, from simple particles in boxes to complex atoms and molecules, bridging the gap between theory and practical applications.

Schrödinger Equation Derivation

Time-Dependent and Time-Independent Forms

Top images from around the web for Time-Dependent and Time-Independent Forms

• 

  Schrödinger equation - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Hamiltonian Simulation by Qubitization – Quantum View original

  Is this image relevant?

• 

  Schrödinger Equation [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Schrödinger equation - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Hamiltonian Simulation by Qubitization – Quantum View original

  Is this image relevant?

1 of 3

Top images from around the web for Time-Dependent and Time-Independent Forms

• 

  Schrödinger equation - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Hamiltonian Simulation by Qubitization – Quantum View original

  Is this image relevant?

• 

  Schrödinger Equation [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Schrödinger equation - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Hamiltonian Simulation by Qubitization – Quantum View original

  Is this image relevant?

1 of 3

• Time-dependent Schrödinger equation describes quantum state evolution over time
  • Fundamental equation in quantum mechanics
  • General form: $i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)$
  • $\Psi(\mathbf{r},t)$ represents the wavefunction
  • $\hat{H}$ denotes the Hamiltonian operator
• Time-independent Schrödinger equation used for systems with time-independent Hamiltonians
  • Derived from the time-dependent equation
  • General form: $\hat{H}\psi(\mathbf{r}) = E\psi(\mathbf{r})$
  • $\psi(\mathbf{r})$ represents the spatial part of the wavefunction
  • $E$ denotes the energy eigenvalue
• Hamiltonian operator represents total energy of the system
  • Crucial in formulating both forms of the Schrödinger equation
  • Typically consists of kinetic and potential energy terms
  • For a particle in a potential: $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})$

Key Concepts and Techniques

• Operators in quantum mechanics represent physical observables mathematically
  • Examples include position ($\hat{x}$), momentum ($\hat{p}$), and energy ($\hat{H}$)
  • Operators act on wavefunctions to produce measurable quantities
• Conservation of energy principle plays key role in deriving time-independent Schrödinger equation
  • Total energy remains constant for conservative systems
  • Allows separation of time and spatial variables
• Separation of variables technique transforms time-dependent equation into time-independent form
  • Assumes wavefunction can be written as product of spatial and temporal parts
  • $\Psi(\mathbf{r},t) = \psi(\mathbf{r})\phi(t)$
  • Leads to two separate equations for spatial and temporal components
• Relationship between classical and quantum mechanics
  • Schrödinger equation reduces to classical equations of motion in the limit of large quantum numbers
  • Correspondence principle connects quantum and classical descriptions (Bohr atomic model)
  • Wave-particle duality (de Broglie wavelength) bridges classical and quantum concepts

Solving the Schrödinger Equation

Fundamental Quantum Systems

• Particle in a box (infinite square well) introduces quantum mechanical concepts
  • One-dimensional potential: $V(x) = \begin{cases} 0 & 0 < x < L \\ \infty & \text{otherwise} \end{cases}$
  • Quantized energy levels: $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$
  • Wavefunctions: $\psi_n(x) = \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L})$
• Harmonic oscillator model crucial for understanding vibrational motion
  • Potential energy: $V(x) = \frac{1}{2}kx^2$
  • Energy levels: $E_n = \hbar\omega(n + \frac{1}{2})$
  • Applications in molecular vibrations and quantum field theory
• Hydrogen atom solution demonstrates application to real physical systems
  • Coulomb potential: $V(r) = -\frac{e^2}{4\pi\epsilon_0r}$
  • Energy levels: $E_n = -\frac{13.6\text{ eV}}{n^2}$
  • Introduces concepts of quantum numbers (n, l, m)

Solution Techniques and Concepts

• Boundary conditions and normalization essential for meaningful solutions
  • Wavefunctions must be continuous and single-valued
  • Normalization ensures total probability equals unity: $\int_{-\infty}^{\infty} |\Psi|^2 d\tau = 1$
• Quantized energy levels emerge naturally from Schrödinger equation solutions
  • Discrete energy states in bound systems (atoms, molecules)
  • Continuous energy spectrum for unbound states (free particles)
• Analytical and numerical methods used for solving Schrödinger equation
  • Analytical solutions available for simple systems (particle in a box, harmonic oscillator)
  • Numerical methods (finite difference, variational) for complex systems
• Symmetry principles simplify solution process for certain quantum systems
  • Parity considerations in symmetric potentials
  • Angular momentum conservation in spherically symmetric problems

Wavefunction Interpretation

Physical Meaning and Properties

• Wavefunction contains all information about quantum system's state
  • Complex-valued function of position and time
  • Represents probability amplitude for various measurable quantities
• Born interpretation relates wavefunction magnitude to probability density
  • Probability density: $P(\mathbf{r},t) = |\Psi(\mathbf{r},t)|^2$
  • Gives probability of finding particle in specific position
• Normalization ensures total probability equals unity
  • Mathematically: $\int_{-\infty}^{\infty} |\Psi(\mathbf{r},t)|^2 d^3r = 1$
  • Physically meaningful wavefunctions must be normalizable
• Wavefunction requirements for physical validity
  • Continuous and single-valued everywhere
  • Square-integrable (finite normalization)
  • Smooth and differentiable (except at potential discontinuities)

Quantum Superposition and Measurement

• Superposition allows quantum states to exist in multiple states simultaneously
  • Linear combination of basis states: $|\Psi\rangle = \sum_i c_i|\psi_i\rangle$
  • Coefficients $c_i$ represent probability amplitudes
• Wavefunction collapse occurs upon measurement
  • System transitions from superposition to definite state
  • Probability of measuring specific state given by |$c_i$|^2
• Wavefunction phase plays crucial role in quantum interference
  • Not directly observable but affects interference patterns
  • Example: double-slit experiment with electrons

Quantum States and Representation

State Vectors and Hilbert Space

• Quantum state provides complete description of quantum system
  • Represented by state vector in Hilbert space
  • Contains all measurable information about system
• Dirac notation (bra-ket notation) standard for representing quantum states
  • Ket |$\psi$⟩ represents state vector
  • Bra ⟨$\psi$| represents dual vector
  • Inner product: ⟨$\phi$|$\psi$⟩
• Pure states represent systems in single, well-defined quantum state
  • Can be described by single state vector
  • Example: electron in specific energy level of hydrogen atom
• Mixed states describe statistical ensembles of pure states
  • Represent systems with classical uncertainty
  • Described using density matrix formalism

Basis States and Quantum Phenomena

• Basis states form complete set of orthonormal vectors
  • Any quantum state expressible as linear combination of basis states
  • Example: energy eigenstates of hydrogen atom
• Degeneracy refers to multiple quantum states sharing same energy level
  • Common in systems with symmetry
  • Example: hydrogen atom with same n but different l quantum numbers
• Coherent states exhibit classical-like behavior
  • Important in quantum optics and laser physics
  • Minimum uncertainty states for position and momentum
• Density matrix formalism describes both pure and mixed states
  • Particularly useful for open quantum systems
  • Represents state as $\rho = \sum_i p_i|\psi_i\rangle\langle\psi_i|$
  • Allows calculation of expectation values and time evolution