You have 3 free guides left 😟
Unlock your guides
You have 3 free guides left 😟
Unlock your guides

The quantum harmonic oscillator builds on classical mechanics, starting with a review of . This foundational concept describes repetitive movement around an equilibrium point, driven by a proportional to displacement.

Understanding the math behind harmonic motion is crucial. Key equations for position, velocity, and acceleration involve , , and phase. These basics set the stage for exploring quantum mechanics' unique take on oscillators.

Fundamentals of Harmonic Motion

Principles of Simple Harmonic Motion

Top images from around the web for Principles of Simple Harmonic Motion
Top images from around the web for Principles of Simple Harmonic Motion
  • Simple harmonic motion describes repetitive back-and-forth movement around an
  • Occurs when a restoring force acts proportionally to the displacement from equilibrium
  • Restoring force always points towards the equilibrium position, opposing the motion of the oscillator
  • Equilibrium position represents the point where the net force on the oscillator is zero
  • Amplitude defines the maximum displacement from the equilibrium position
  • Motion follows a sinusoidal pattern over time, represented by sine or cosine functions
  • Examples include pendulums swinging (grandfather clocks) and masses on springs (vehicle suspension systems)

Mathematical Representation of Harmonic Motion

  • Position as a function of time: x(t)=Acos(ωt+ϕ)x(t) = A \cos(\omega t + \phi)
  • Velocity as a function of time: v(t)=Aωsin(ωt+ϕ)v(t) = -A\omega \sin(\omega t + \phi)
  • Acceleration as a function of time: a(t)=Aω2cos(ωt+ϕ)a(t) = -A\omega^2 \cos(\omega t + \phi)
  • AA represents the amplitude, ω\omega the angular frequency, and ϕ\phi the phase constant
  • These equations describe the motion of an ideal harmonic oscillator without damping or external forces
  • Real-world examples often involve some degree of damping (air resistance in pendulums)

Oscillation Characteristics

Frequency and Period Relationships

  • Frequency (ff) measures the number of oscillations per unit time, typically expressed in Hertz (Hz)
  • Period (TT) represents the time required for one complete oscillation
  • Inverse relationship between frequency and period: T=1fT = \frac{1}{f}
  • Higher frequency corresponds to shorter periods (tuning forks with high pitch)
  • Lower frequency corresponds to longer periods (ocean waves with long wavelengths)
  • Natural frequency depends on the system's physical properties (mass and spring constant for a )

Angular Frequency and Its Significance

  • Angular frequency (ω\omega) measures the rate of angular displacement in radians per second
  • Relates to linear frequency: ω=2πf=2πT\omega = 2\pi f = \frac{2\pi}{T}
  • Appears in the equations of motion for harmonic oscillators
  • Determines the speed of oscillation and energy exchange rate between kinetic and potential forms
  • Influences the resonance behavior of forced oscillators (radio tuning circuits)
  • Can be manipulated in engineered systems to achieve desired oscillation characteristics (vehicle shock absorbers)

Energy in Harmonic Oscillators

Hooke's Law and Potential Energy

  • describes the restoring force in ideal springs: F=kxF = -kx
  • kk represents the spring constant, measuring the spring's stiffness
  • Negative sign indicates the force opposes the displacement
  • of a harmonic oscillator: U=12kx2U = \frac{1}{2}kx^2
  • Potential energy reaches maximum at the extremes of motion (fully compressed or extended spring)
  • Varies parabolically with displacement, creating a symmetric potential well
  • Examples include guitar strings and diving boards storing elastic potential energy

Kinetic Energy and Energy Conservation

  • of a harmonic oscillator: K=12mv2K = \frac{1}{2}mv^2
  • mm represents the mass of the oscillating object
  • Kinetic energy reaches maximum at the equilibrium position ( at lowest point)
  • Total energy of the system remains constant in ideal harmonic motion: E=K+U=constantE = K + U = \text{constant}
  • Energy continuously converts between potential and kinetic forms throughout the oscillation
  • Conservation of energy principle allows prediction of motion characteristics
  • Real-world applications include energy harvesting from vibrations (piezoelectric devices)
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Glossary
Glossary