Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Quantum mechanics ↓


Quantum States


Understanding quantum states is a fundamental aspect of quantum mechanics, the branch of physics that deals with the strange and fascinating behavior of matter and light at the atomic and subatomic scale. A quantum state describes the state of a quantum system - it can be an atom, a particle, or any other quantum entity. In this detailed explanation, we will unravel the mysteries of quantum states and discuss their properties, visual representations, examples, and their impact on the physical world.

What are quantum states?

The quantum state gives us all the information we can in principle know about a quantum system. Unlike classical states, which give precise information about the characteristics of a system, quantum states obey principles of quantum mechanics such as superposition and uncertainty. These principles imply that we can only talk about the probabilities of finding a quantum system in a particular state when making a measurement.

In the formal language of quantum mechanics, quantum states are represented by vectors in a complex vector space called a Hilbert space. State vectors (or ket vectors, denoted by |ψ>) can be used to contain this information.

Superposition principle

One of the most interesting aspects of quantum states is the principle of superposition. According to this principle, if a quantum system can be in any one of several different states, it can also exist in a combination, or superposition, of these states. Superposition is different from anything found in classical physics, because it allows for the possibility of being in multiple states at once.

For example, consider an electron that can exist in either state |A> or state |B>. According to the superposition principle, the electron can also exist in the following states:

|ψ> = c1|A> + c2|B>

where c1 and c2 are complex coefficients that determine the probability of the electron being found in state |A> or state |B> after the measurement. The probabilities are given by the squares of the magnitudes of these coefficients, |c1|2 and |c2|2, and must sum to 1.

Examples of quantum states

Let us consider some simple but informative examples of quantum states to understand how they manifest in real systems.

Example 1: Spin-1/2 particle

Spin is a fundamental property of particles, just like mass or charge. Particles with 1/2 spin, such as electrons, can exist in two possible states, often called spin-up and spin-down. In Dirac notation, these states are represented as:

|↑> = |1/2, +>
|↓> = |1/2, ->

A general quantum state for a spin-1/2 particle can be written as a superposition of these two states:

|ψ> = α|↑> + β|↓>

Here, α and β are complex numbers satisfying |α|2 + |β|2 = 1.

Example 2: Quantum harmonic oscillator

A quantum harmonic oscillator is a model that describes oscillating particles in quantum mechanics, just like a mass on a spring in classical mechanics. The energy levels of a quantum harmonic oscillator are quantized, meaning they can only take on specific discrete values. These quantized energy states are represented as:

|n>

where n = 0, 1, 2, ... denotes the quantum number associated with each state. Each state |n> has a characteristic energy:

En = ℏω(n + 1/2)

where ℏ is the reduced Planck constant and ω is the angular frequency of the oscillator.

Visualization of quantum states

Vector representation

A common method for visualizing quantum states is to use the block field for a two-level quantum system, much like a spin-1/2 particle system.

Consider the state:

|ψ> = α|0> + β|1>

This can be visualized on a block sphere, where any point on the surface of the sphere corresponds to a particular quantum state. The state |0> is at the north pole, |1> is at the south pole, and any superposition is a point on the surface.

|0> |1>

In this representation, any quantum state can be represented as a vector pointing to a specific location on the surface of the sphere.

Wave function representation

Another way to represent quantum states is through the wave function, usually represented by the Greek letter psi (ψ). The wave function captures the probability amplitude of a quantum system in space and time. For a particle in a one-dimensional box, the wave function can look like sinusoidal waves.

For example, for a particle in a one-dimensional box of length L, the allowed wave functions are:

ψn(x) = sqrt(2/L) sin(nπx/L)

where n is the quantum number, indicating the different energy states. Each of these wave functions corresponds to a specific eigenstate of the system.

Quantum measurement and collapse of the quantum state

In quantum mechanics, measurement is an important process that affects the state of a system. When we measure a quantum system, the act of measurement causes the wave function to 'collapse' into one of the eigenstates of the measurement operator. This collapse is a probabilistic phenomenon, governed by the statistical interpretation of the wave function.

For example, consider measuring the spin state of an electron initially in superposition:

|ψ> = α|↑> + β|↓>

The measurement will result in the electron being found to be in either the spin-up state |↑> or the spin-down state |↓>, with probabilities |α|2 and |β|2 respectively.

Conclusion

The concept of quantum states is a cornerstone of quantum mechanics and sheds light on the non-intuitive nature of quantum phenomena. Quantum states, through superposition, give rise to the unique properties of quantum mechanics, including the emergence of quantum entanglement and interference phenomena. The probabilistic nature of quantum states and the principle of measurement add to the fascinating complexity residues of the quantum world.

As we advance in the study of quantum mechanics, concepts such as quantum states become increasingly essential – forming the building blocks of further technologies, such as quantum computing, quantum cryptography, and advanced quantum simulation.


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