A simple harmonic oscillator is an oscillator that is neither driven nor damped.It consists of a mass m, which experiences a single force F, which pulls the mass in the direction of the point x = 0 and depends only on the mass's position x and a constant k.
Some trajectories of a harmonic oscillator according to Newton's laws of classical mechanics (AâB), and according to the Schrödinger equation of quantum mechanics (CâH). In AâB, the particle (represented as a ball attached to a spring) oscillates back and forth. In CâH, some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. C, D, E, F, but not G, H, are energy eigenstates. H is a coherent stateâa quantum state that approximates the classical trajectory.
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.[1][2][3]
One-dimensional harmonic oscillator[edit]Hamiltonian and energy eigenstates[edit]
Wavefunction representations for the first eight bound eigenstates, n = 0 to 7. The horizontal axis shows the position x. Note: The graphs are not normalized, and the signs of some of the functions differ from those given in the text.
Corresponding probability densities.
The Hamiltonian of the particle is:
where m is the particle's mass, k is the force constant, Ï=km{displaystyle omega ={sqrt {frac {k}{m}}}} is the angular frequency of the oscillator, x^{displaystyle {hat {x}}} is the position operator (given by x), and p^{displaystyle {hat {p}}} is the momentum operator (given by p^=âiâââx{displaystyle {hat {p}}=-ihbar {partial over partial x},}). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law.
One may write the time-independent Schrödinger equation,
where E denotes a to-be-determined real number that will specify a time-independent energy level, or eigenvalue, and the solution |Ïâ© denotes that level's energy eigenstate.
One may solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave functionâ¨x|Ïâ© = Ï(x), using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions,
The functions Hn are the physicists' Hermite polynomials,
The corresponding energy levels are
This energy spectrum is noteworthy for three reasons. First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of ħÏ) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy (the energy of the n = 0 state, called the ground state) is not equal to the minimum of the potential well, but ħÏ/2 above it; this is called zero-point energy. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in a classical oscillator), but have a small range of variance, in accordance with the Heisenberg uncertainty principle.
The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density peaks at the classical 'turning points', where the state's energy coincides with the potential energy. (See the discussion below of the highly excited states.) This is consistent with the classical harmonic oscillator, in which the particle spends more of its time (and is therefore more likely to be found) near the turning points, where it is moving the slowest. The correspondence principle is thus satisfied. Moreover, special nondispersive wave packets, with minimum uncertainty, called coherent states oscillate very much like classical objects, as illustrated in the figure; they are not eigenstates of the Hamiltonian.
Ladder operator method[edit]
Probability densities |Ïn(x)|2 for the bound eigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position x, and brighter colors represent higher probability densities.
The 'ladder operator' method, developed by Paul Dirac, allows extraction of the energy eigenvalues without directly solving the differential equation. It is generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators a and its adjointaâ ,
This leads to the useful representation of x^{displaystyle {hat {x}}} and p^{displaystyle {hat {p}}},
The operator a is not Hermitian, since itself and its adjoint aâ are not equal. The energy eigenstates |nâ©, when operated on by these ladder operators, give
It is then evident that aâ , in essence, appends a single quantum of energy to the oscillator, while a removes a quantum. For this reason, they are sometimes referred to as 'creation' and 'annihilation' operators.
From the relations above, we can also define a number operator N, which has the following property:
The following commutators can be easily obtained by substituting the canonical commutation relation,
And the Hamilton operator can be expressed as
so the eigenstate of N is also the eigenstate of energy.
The commutation property yields
and similarly,
This means that a acts on |nâ© to produce, up to a multiplicative constant, |nâ1â©, and aâ acts on |nâ© to produce |n+1â©. For this reason, a is called a annihilation operator ('lowering operator'), and aâ a creation operator ('raising operator'). The two operators together are called ladder operators. In quantum field theory, a and aâ are alternatively called 'annihilation' and 'creation' operators because they destroy and create particles, which correspond to our quanta of energy.
Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with Ä§Ï less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to E = ââ. However, since
the smallest eigen-number is 0, and
In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstates. Furthermore, we have shown above that
Finally, by acting on |0â© with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates
such that
which matches the energy spectrum given in the preceding section.
Arbitrary eigenstates can be expressed in terms of |0â©,
Analytical questions[edit]
The preceding analysis is algebraic, using only the commutation relations between the raising and lowering operators. Once the algebraic analysis is complete, one should turn to analytical questions. First, one should find the ground state, that is, the solution of the equation aÏ0=0{displaystyle apsi _{0}=0}. In the position representation, this is the first-order differential equation
whose solution is easily found to be the Gaussian[4]
Conceptually, it is important that there is only one solution of this equation; if there were, say, two linearly independent ground states, we would get two independent chains of eigenvectors for the harmonic oscillator. Once the ground state is computed, one can show inductively that the excited states are Hermite polynomials times the Gaussian ground state, using the explicit form of the raising operator in the position representation. One can also prove that, as expected from the uniqueness of the ground state, the energy eigenstates Ïn{displaystyle psi _{n}} constructed by the ladder method form a complete orthonormal set of functions.[5]
Explicitly connecting with the previous section, the ground state |0â© in the position representation is determined by a|0â©=0{displaystyle a|0rangle =0},
hence
so that Ï1(x,t)=â¨xâ£eâ3iÏt/2aâ â£0â©{displaystyle psi _{1}(x,t)=langle xmid e^{-3iomega t/2}a^{dagger }mid 0rangle }, and so on.
Natural length and energy scales[edit]
The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. These can be found by nondimensionalization.
The result is that, if energy is measured in units of Ä§Ï and distance in units of âħ/(mÏ), then the Hamiltonian simplifies to
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while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half,
where Hn(x) are the Hermite polynomials.
To avoid confusion, these 'natural units' will mostly not be adopted in this article. However, they frequently come in handy when performing calculations, by bypassing clutter.
For example, the fundamental solution (propagator) of Hâiât, the time-dependent Schrödinger operator for this oscillator, simply boils down to the Mehler kernel,[6][7]
where K(x,y;0) =δ(x â y). The most general solution for a given initial configuration Ï(x,0) then is simply
Coherent states[edit]
The coherent states of the harmonic oscillator are special nondispersive wave packets, with minimum uncertainty Ïx Ïp = âââ2, whose observables' expectation values evolve like a classical system. They are eigenvectors of the annihilation operator, not the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality.
The coherent states are indexed by α â â and expressed in the |nâ© basis as
Because a|0â©=0{displaystyle aleft|0rightrangle =0} and via the Kermack-McCrae identity, the last form is equivalent to a unitarydisplacement operator acting on the ground state: |αâ©=eαa^â âαâa^|0â©=D(α)|0â©{displaystyle |alpha rangle =e^{alpha {hat {a}}^{dagger }-alpha ^{*}{hat {a}}}|0rangle =D(alpha )|0rangle }. The position space wave functions are
Highly excited states[edit]
Excited state with n=30, with the vertical lines indicating the turning points
When n is large, the eigenstates are localized into the classical allowed region, that is, the region in which a classical particle with energy En can move. The eigenstates are peaked near the turning points: the points at the ends of the classically allowed region where the classical particle changes direction. This phenomenon can be verified through asymptotics of the Hermite polynomials, and also through the WKB approximation.
The frequency of oscillation at x is proportional to the momentum p(x) of a classical particle of energy En and position x. Furthermore, the square of the amplitude (determining the probability density) is inversely proportional to p(x), reflecting the length of time the classical particle spends near x. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an Airy function. Using properties of the Airy function, one may estimate the probability of finding the particle outside the classically allowed region, to be approximately
This is also given, asymptotically, by the integral
Phase space solutions[edit]
In the phase space formulation of quantum mechanics, solutions to the quantum harmonic oscillator in several different representations of the quasiprobability distribution can be written in closed form. The most widely used of these is for the Wigner quasiprobability distribution, which has the solution
where
and Ln are the Laguerre polynomials.
This example illustrates how the Hermite and Laguerre polynomials are linked through the Wigner map.
Meanwhile, the Husimi Q function of the harmonic oscillator eigenstates have an even simpler form. If we work in the natural units described above, we have
This claim can be verified using the SegalâBargmann transform. Specifically, since the raising operator in the SegalâBargmann representation is simply multiplication by z=x+ip{displaystyle z=x+ip} and the ground state is the constant function 1, the normalized harmonic oscillator states in this representation are simply zn/n!{displaystyle z^{n}/{sqrt {n!}}} . At this point, we can appeal to the formula for the Husimi Q function in terms of the SegalâBargmann transform.
N-dimensional harmonic oscillator[edit]
The one-dimensional harmonic oscillator is readily generalizable to N dimensions, where N = 1, 2, 3, .. . In one dimension, the position of the particle was specified by a single coordinate, x. In N dimensions, this is replaced by N position coordinates, which we label x1, .., xN. Corresponding to each position coordinate is a momentum; we label these p1, .., pN. The canonical commutation relations between these operators are
The Hamiltonian for this system is
As the form of this Hamiltonian makes clear, the N-dimensional harmonic oscillator is exactly analogous to N independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantities x1, .., xN would refer to the positions of each of the N particles. This is a convenient property of the r2{displaystyle r^{2}} potential, which allows the potential energy to be separated into terms depending on one coordinate each.
This observation makes the solution straightforward. For a particular set of quantum numbers {n} the energy eigenfunctions for the N-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:
In the ladder operator method, we define N sets of ladder operators,
By an analogous procedure to the one-dimensional case, we can then show that each of the ai and aâ i operators lower and raise the energy by âÏ respectively. The Hamiltonian is
This Hamiltonian is invariant under the dynamic symmetry group U(N) (the unitary group in N dimensions), defined by
where Uji{displaystyle U_{ji}} is an element in the defining matrix representation of U(N).
The energy levels of the system are
As in the one-dimensional case, the energy is quantized. The ground state energy is N times the one-dimensional ground energy, as we would expect using the analogy to N independent one-dimensional oscillators. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. In N-dimensions, except for the ground state, the energy levels are degenerate, meaning there are several states with the same energy.
The degeneracy can be calculated relatively easily. As an example, consider the 3-dimensional case: Define n = n1 + n2 + n3. All states with the same n will have the same energy. For a given n, we choose a particular n1. Then n2 + n3 = n â n1. There are n â n1 + 1 possible pairs {n2, n3}. n2 can take on the values 0 to n â n1, and for each n2 the value of n3 is fixed. The degree of degeneracy therefore is:
Formula for general N and n [gn being the dimension of the symmetric irreducible nth power representation of the unitary group U(N)]:
The special case N = 3, given above, follows directly from this general equation. This is however, only true for distinguishable particles, or one particle in N dimensions (as dimensions are distinguishable). For the case of N bosons in a one-dimension harmonic trap, the degeneracy scales as the number of ways to partition an integer n using integers less than or equal to N.
This arises due to the constraint of putting N quanta into a state ket where âk=0âknk=n{displaystyle sum _{k=0}^{infty }kn_{k}=n} and âk=0ânk=N{displaystyle sum _{k=0}^{infty }n_{k}=N}, which are the same constraints as in integer partition.
Example: 3D isotropic harmonic oscillator[edit]
Schrödinger 3D spherical harmonic orbital solutions in 2D density plots; the Mathematica source code that used for generating the plots is at the top
The Schrödinger equation of a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables; see this article for the present case. This procedure is analogous to the separation performed in the hydrogen-like atom problem, but with the spherically symmetric potential
where μ is the mass of the problem. Because m will be used below for the magnetic quantum number, mass is indicated by μ, instead of m, as earlier in this article.
The solution reads[8]
where
are generalized Laguerre polynomials; The order k of the polynomial is a non-negative integer;
The energy eigenvalue is
The energy is usually described by the single quantum number
Because k is a non-negative integer, for every even n we have â = 0, 2, .., n â 2, n and for every odd n we have â = 1, 3, .., n â 2, n . The magnetic quantum number m is an integer satisfying ââ ⤠m ⤠â, so for every n and â there are 2â + 1 different quantum states, labeled by m . Thus, the degeneracy at level n is
where the sum starts from 0 or 1, according to whether n is even or odd.This result is in accordance with the dimension formula above, and amounts to the dimensionality of a symmetric representation of SU(3),[9] the relevant degeneracy group.
Applications[edit]Harmonic oscillators lattice: phonons[edit]
We can extend the notion of a harmonic oscillator to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions.
As in the previous section, we denote the positions of the masses by x1,x2,.., as measured from their equilibrium positions (i.e. xi = 0 if the particle i is at its equilibrium position). In two or more dimensions, the xi are vector quantities. The Hamiltonian for this system is
where m is the (assumed uniform) mass of each atom, and xi and pi are the position and momentum operators for the i th atom and the sum is made over the nearest neighbors (nn). However, it is customary to rewrite the Hamiltonian in terms of the normal modes of the wavevector rather than in terms of the particle coordinates so that one can work in the more convenient Fourier space.
We introduce, then, a set of N 'normal coordinates' Qk, defined as the discrete Fourier transforms of the xs, and N 'conjugate momenta' Î defined as the Fourier transforms of the ps,
The quantity kn will turn out to be the wave number of the phonon, i.e. 2Ï divided by the wavelength. It takes on quantized values, because the number of atoms is finite.
This preserves the desired commutation relations in either real space or wave vector space
From the general result
it is easy to show, through elementary trigonometry, that the potential energy term is
where
The Hamiltonian may be written in wave vector space as
Note that the couplings between the position variables have been transformed away; if the Qs and Î s were hermitian(which they are not), the transformed Hamiltonian would describe Nuncoupled harmonic oscillators.
The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the (N + 1)th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is
The upper bound to n comes from the minimum wavelength, which is twice the lattice spacing a, as discussed above.
The harmonic oscillator eigenvalues or energy levels for the mode Ïk are
If we ignore the zero-point energy then the levels are evenly spaced at
So an exact amount of energyħÏ, must be supplied to the harmonic oscillator lattice to push it to the next energy level. In comparison to the photon case when the electromagnetic field is quantised, the quantum of vibrational energy is called a phonon.
All quantum systems show wave-like and particle-like properties. The particle-like properties of the phonon are best understood using the methods of second quantization and operator techniques described later.[10]
In the continuum limit, aâ0, Nââ, while Na is held fixed. The canonical coordinates Qk devolve to the decoupled momentum modes of a scalar field, Ïk{displaystyle phi _{k}}, whilst the location index i (not the displacement dynamical variable) becomes the parameter x argument of the scalar field, Ï(x,t){displaystyle phi (x,t)}.
Molecular vibrations[edit]
See also[edit]References[edit]
External links[edit]
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Quantum_harmonic_oscillator&oldid=917287252'
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring forceFproportional to the displacement x:
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidaloscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.
If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.
Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.
Simple harmonic oscillator[edit]
Mass-spring harmonic oscillator
Simple harmonic motion
A simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a mass m, which experiences a single force F, which pulls the mass in the direction of the point x = 0 and depends only on the mass's position x and a constant k. Balance of forces (Newton's second law) for the system is
Solving this differential equation, we find that the motion is described by the function
where
The motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude A. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its periodT=2Ï/Ï{displaystyle T=2pi /omega }, the time for a single oscillation or its frequency f=1/T{displaystyle f=1/T}, the number of cycles per unit time. The position at a given time t also depends on the phaseÏ, which determines the starting point on the sine wave. The period and frequency are determined by the size of the mass m and the force constant k, while the amplitude and phase are determined by the starting position and velocity.
The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases. The velocity is maximal for zero displacement, while the acceleration is in the direction opposite to the displacement.
The potential energy stored in a simple harmonic oscillator at position x is
Damped harmonic oscillator[edit]
Dependence of the system behavior on the value of the damping ratio ζ
Video clip demonstrating a damped harmonic oscillator consisting of a dynamics cart between two springs. An accelerometer on top of the cart shows the magnitude and direction of the acceleration.
In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases in proportion to the acting frictional force. While in a simple undriven harmonic oscillator the only force acting on the mass is the restoring force, in a damped harmonic oscillator there is in addition a frictional force which is always in a direction to oppose the motion. In many vibrating systems the frictional force Ff can be modeled as being proportional to the velocity v of the object: Ff = âcv, where c is called the viscous damping coefficient.
The balance of forces (Newton's second law) for damped harmonic oscillators is then
which can be rewritten into the form
where
Step response of a damped harmonic oscillator; curves are plotted for three values of μ = Ï1 = Ï0â1 â ζ2. Time is in units of the decay time Ï = 1/(ζÏ0).
The value of the damping ratio ζ critically determines the behavior of the system. A damped harmonic oscillator can be:
The Q factor of a damped oscillator is defined as
Q is related to the damping ratio by the equation Q=12ζ.{displaystyle Q={frac {1}{2zeta }}.}
Driven harmonic oscillators[edit]
Driven harmonic oscillators are damped oscillators further affected by an externally applied force F(t).
Newton's second law takes the form
It is usually rewritten into the form
This equation can be solved exactly for any driving force, using the solutions z(t) that satisfy the unforced equation
and which can be expressed as damped sinusoidal oscillations:
in the case where ζ ⤠1. The amplitude A and phase Ï determine the behavior needed to match the initial conditions.
Step input[edit]
In the case ζ < 1 and a unit step input with x(0) = 0:
the solution is
with phase Ï given by
The time an oscillator needs to adapt to changed external conditions is of the order Ï = 1/(ζÏ0). In physics, the adaptation is called relaxation, and Ï is called the relaxation time.
In electrical engineering, a multiple of Ï is called the settling time, i.e. the time necessary to ensure the signal is within a fixed departure from final value, typically within 10%. The term overshoot refers to the extent the response maximum exceeds final value, and undershoot refers to the extent the response falls below final value for times following the response maximum.
Sinusoidal driving force[edit]
Steady-state variation of amplitude with relative frequency Ï/Ï0{displaystyle omega /omega _{0}} and dampingζ{displaystyle zeta } of a driven simple harmonic oscillator
In the case of a sinusoidal driving force:
where F0{displaystyle F_{0}} is the driving amplitude, and Ï{displaystyle omega } is the driving frequency for a sinusoidal driving mechanism. This type of system appears in AC-driven RLC circuits (resistorâinductorâcapacitor) and driven spring systems having internal mechanical resistance or external air resistance.
The general solution is a sum of a transient solution that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on the driving amplitude F0{displaystyle F_{0}}, driving frequency Ï{displaystyle omega }, undamped angular frequency Ï0{displaystyle omega _{0}}, and the damping ratio ζ{displaystyle zeta }.
The steady-state solution is proportional to the driving force with an induced phase change Ï{displaystyle varphi }:
where
is the absolute value of the impedance or linear response function, and
is the phase of the oscillation relative to the driving force. The phase value is usually taken to be between â180° and 0 (that is, it represents a phase lag, for both positive and negative values of the arctan argument).
For a particular driving frequency called the resonance, or resonant frequency Ïr=Ï01â2ζ2{displaystyle omega _{r}=omega _{0}{sqrt {1-2zeta ^{2}}}}, the amplitude (for a given F0{displaystyle F_{0}}) is maximal. This resonance effect only occurs when ζ<1/2{displaystyle zeta <1/{sqrt {2}}}, i.e. for significantly underdamped systems. For strongly underdamped systems the value of the amplitude can become quite large near the resonant frequency.
The transient solutions are the same as the unforced (F0=0{displaystyle F_{0}=0}) damped harmonic oscillator and represent the systems response to other events that occurred previously. The transient solutions typically die out rapidly enough that they can be ignored.
Parametric oscillators[edit]
A parametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force.A familiar example of parametric oscillation is 'pumping' on a playground swing.[4][5][6]A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ('pumping') or alternately standing and squatting, in rhythm with the swing's oscillations. The varying of the parameters drives the system. Examples of parameters that may be varied are its resonance frequency Ï{displaystyle omega } and damping β{displaystyle beta }.
Parametric oscillators are used in many applications. The classical varactor parametric oscillator oscillates when the diode's capacitance is varied periodically. The circuit that varies the diode's capacitance is called the 'pump' or 'driver'. In microwave electronics, waveguide/YAG based parametric oscillators operate in the same fashion. The designer varies a parameter periodically to induce oscillations.
Parametric oscillators have been developed as low-noise amplifiers, especially in the radio and microwave frequency range. Thermal noise is minimal, since a reactance (not a resistance) is varied. Another common use is frequency conversion, e.g., conversion from audio to radio frequencies. For example, the Optical parametric oscillator converts an input laser wave into two output waves of lower frequency (Ïs,Ïi{displaystyle omega _{s},omega _{i}}).
Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing, since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits the instability phenomenon.
Universal oscillator equation[edit]
The equation
is known as the universal oscillator equation, since all second-order linear oscillatory systems can be reduced to this form.[citation needed] This is done through nondimensionalization.
Simple Harmonic Oscillator Equation Proof
If the forcing function is f(t) = cos(Ït) = cos(ÏtcÏ) = cos(ÏÏ), where Ï = Ïtc, the equation becomes
Simple Harmonic Oscillator Pdf Free
The solution to this differential equation contains two parts: the 'transient' and the 'steady-state'.
Transient solution[edit]
The solution based on solving the ordinary differential equation is for arbitrary constants c1 and c2
The transient solution is independent of the forcing function.
Steady-state solution[edit]
Apply the 'complex variables method' by solving the auxiliary equation below and then finding the real part of its solution:
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Supposing the solution is of the form
Its derivatives from zeroth to second order are
Pc games. Substituting these quantities into the differential equation gives
Dividing by the exponential term on the left results in
Equating the real and imaginary parts results in two independent equations
Amplitude part[edit]
Bode plot of the frequency response of an ideal harmonic oscillator
Squaring both equations and adding them together gives
Therefore,
Compare this result with the theory section on resonance, as well as the 'magnitude part' of the RLC circuit. This amplitude function is particularly important in the analysis and understanding of the frequency response of second-order systems.
Phase part[edit]
To solve for Ï, divide both equations to get
This phase function is particularly important in the analysis and understanding of the frequency response of second-order systems.
Full solution[edit]
Combining the amplitude and phase portions results in the steady-state solution
The solution of original universal oscillator equation is a superposition (sum) of the transient and steady-state solutions:
For a more complete description of how to solve the above equation, see linear ODEs with constant coefficients.
Equivalent systems[edit]
Harmonic oscillators occurring in a number of areas of engineering are equivalent in the sense that their mathematical models are identical (see universal oscillator equation above). Below is a table showing analogous quantities in four harmonic oscillator systems in mechanics and electronics. If analogous parameters on the same line in the table are given numerically equal values, the behavior of the oscillators â their output waveform, resonant frequency, damping factor, etc. â are the same.
Application to a conservative force[edit]
The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of any conservative force, in the limit of small motions, behaves as a simple harmonic oscillator.
A conservative force is one that is associated with a potential energy. The potential-energy function of a harmonic oscillator is
Harmonic Oscillator Solution
Given an arbitrary potential-energy function V(x){displaystyle V(x)}, one can do a Taylor expansion in terms of x{displaystyle x} around an energy minimum (x=x0{displaystyle x=x_{0}}) to model the behavior of small perturbations from equilibrium.
Because V(x0){displaystyle V(x_{0})} is a minimum, the first derivative evaluated at x0{displaystyle x_{0}} must be zero, so the linear term drops out:
The constant termV(x0) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:
Thus, given an arbitrary potential-energy function V(x){displaystyle V(x)} with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.
Examples[edit]Simple pendulum[edit]
A simple pendulum exhibits approximately simple harmonic motion under the conditions of no damping and small amplitude.
Assuming no damping, the differential equation governing a simple pendulum of length l{displaystyle l}, where g{displaystyle g} is the local acceleration of gravity, is
If the maximal displacement of the pendulum is small, we can use the approximation sinâ¡Î¸âθ{displaystyle sin theta approx theta } and instead consider the equation
The general solution to this differential equation is
where A{displaystyle A} and Ï{displaystyle varphi } are constants that depends on the initial conditions.Using as initial conditions θ(0)=θ0{displaystyle theta (0)=theta _{0}} and θË(0)=0{displaystyle {dot {theta }}(0)=0}, the solution is given by
where θ0{displaystyle theta _{0}} is the largest angle attained by the pendulum (that is, θ0{displaystyle theta _{0}} is the amplitude of the pendulum). The period, the time for one complete oscillation, is given by the expression
which is a good approximation of the actual period when θ0{displaystyle theta _{0}} is small. Notice that in this approximation the period Ï{displaystyle tau } is independent of the amplitude θ0{displaystyle theta _{0}}. In the above equation, Ï{displaystyle omega } represents the angular frequency.
Spring/mass system[edit]
Springâmass system in equilibrium (A), compressed (B) and stretched (C) states
When a spring is stretched or compressed by a mass, the spring develops a restoring force. Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length:
where F is the force, k is the spring constant, and x is the displacement of the mass with respect to the equilibrium position. The minus sign in the equation indicates that the force exerted by the spring always acts in a direction that is opposite to the displacement (i.e. the force always acts towards the zero position), and so prevents the mass from flying off to infinity.
By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation:
the latter being Newton's second law of motion.
If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by
Given an ideal massless spring, m{displaystyle m} is the mass on the end of the spring. If the spring itself has mass, its effective mass must be included in m{displaystyle m}.
Energy variation in the springâdamping system[edit]
In terms of energy, all systems have two types of energy: potential energy and kinetic energy. When a spring is stretched or compressed, it stores elastic potential energy, which then is transferred into kinetic energy. The potential energy within a spring is determined by the equation U=kx2/2.{displaystyle U=kx^{2}/2.}
When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximal potential energy, the kinetic energy of the mass is zero. When the spring is released, it tries to return to equilibrium, and all its potential energy converts to kinetic energy of the mass.
Definition of terms[edit]
See also[edit]Notes[edit]
References[edit]
External links[edit]
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