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Third, the probability density distributions | ψ n ( x ) | 2 for a quantum oscillator in the ground low-energy state, ψ 0 ( x ) , is largest at the middle of the well ( x = 0 ) . For the particle to be found with greatest probability at the center of the well, we expect that the particle spends the most time there as it oscillates. This is opposite to the behavior of a classical oscillator, in which the particle spends most of its time moving with relative small speeds near the turning points.

Check Your Understanding Find the expectation value of the position for a particle in the ground state of a harmonic oscillator using symmetry.

x = 0

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Quantum probability density distributions change in character for excited states, becoming more like the classical distribution when the quantum number gets higher. We observe this change already for the first excited state of a quantum oscillator because the distribution | ψ 1 ( x ) | 2 peaks up around the turning points and vanishes at the equilibrium position, as seen in [link] . In accordance with Bohr’s correspondence principle, in the limit of high quantum numbers, the quantum description of a harmonic oscillator converges to the classical description, which is illustrated in [link] . The classical probability density distribution corresponding to the quantum energy of the n = 12 state is a reasonably good approximation of the quantum probability distribution for a quantum oscillator in this excited state. This agreement becomes increasingly better for highly excited states.

The probability density distribution amplitude squared of Psi sub 12 for the quantum harmonic oscillator is plotted as a function of x as a solid curve. The curve has 13 peaks with 12 zeros between them and goes asymptotically to zero at plus and minus infinity. The amplitude of the peaks is lowest at the center and increases wit distance from the origin. All of the peaks are between x=-A and x=+A. The dashed curve which shows the probability density distribution of a classical oscillator with the same energy is a smooth upward opening curve.
The probability density distribution for finding the quantum harmonic oscillator in its n = 12 quantum state. The dashed curve shows the probability density distribution of a classical oscillator with the same energy.

Summary

  • The quantum harmonic oscillator is a model built in analogy with the model of a classical harmonic oscillator. It models the behavior of many physical systems, such as molecular vibrations or wave packets in quantum optics.
  • The allowed energies of a quantum oscillator are discrete and evenly spaced. The energy spacing is equal to Planck’s energy quantum.
  • The ground state energy is larger than zero. This means that, unlike a classical oscillator, a quantum oscillator is never at rest, even at the bottom of a potential well, and undergoes quantum fluctuations.
  • The stationary states (states of definite energy) have nonzero values also in regions beyond classical turning points. When in the ground state, a quantum oscillator is most likely to be found around the position of the minimum of the potential well, which is the least-likely position for a classical oscillator.
  • For high quantum numbers, the motion of a quantum oscillator becomes more similar to the motion of a classical oscillator, in accordance with Bohr’s correspondence principle.

Conceptual questions

Is it possible to measure energy of 0.75 ω for a quantum harmonic oscillator? Why? Why not? Explain.

No. This energy corresponds to n = 0.25 , but n must be an integer.

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Explain the connection between Planck’s hypothesis of energy quanta and the energies of the quantum harmonic oscillator.

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If a classical harmonic oscillator can be at rest, why can the quantum harmonic oscillator never be at rest? Does this violate Bohr’s correspondence principle?

Because the smallest allowed value of the quantum number n for a simple harmonic oscillator is 0. No, because quantum mechanics and classical mechanics agree only in the limit of large n .

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Use an example of a quantum particle in a box or a quantum oscillator to explain the physical meaning of Bohr’s correspondence principle.

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Can we simultaneously measure position and energy of a quantum oscillator? Why? Why not?

Yes, within the constraints of the uncertainty principle. If the oscillating particle is localized, the momentum and therefore energy of the oscillator are distributed.

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Problems

Show that the two lowest energy states of the simple harmonic oscillator, ψ 0 ( x ) and ψ 1 ( x ) from [link] , satisfy [link] .

proof

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If the ground state energy of a simple harmonic oscillator is 1.25 eV, what is the frequency of its motion?

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When a quantum harmonic oscillator makes a transition from the ( n + 1 ) state to the n state and emits a 450-nm photon, what is its frequency?

6.662 × 10 14 Hz

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Vibrations of the hydrogen molecule H 2 can be modeled as a simple harmonic oscillator with the spring constant k = 1.13 × 10 3 N / m and mass m = 1.67 × 10 −27 kg . (a) What is the vibrational frequency of this molecule? (b) What are the energy and the wavelength of the emitted photon when the molecule makes transition between its third and second excited states?

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A particle with mass 0.030 kg oscillates back-and-forth on a spring with frequency 4.0 Hz. At the equilibrium position, it has a speed of 0.60 m/s. If the particle is in a state of definite energy, find its energy quantum number.

n 2.037 × 10 30

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Find the expectation value x 2 of the square of the position for a quantum harmonic oscillator in the ground state. Note: + d x x 2 e a x 2 = π ( 2 a 3 / 2 ) 1 .

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Determine the expectation value of the potential energy for a quantum harmonic oscillator in the ground state. Use this to calculate the expectation value of the kinetic energy.

x = 0.5 m ω 2 x 2 = ω / 4 ; K = E U = ω / 4

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Verify that ψ 1 ( x ) given by [link] is a solution of Schrӧdinger’s equation for the quantum harmonic oscillator.

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Estimate the ground state energy of the quantum harmonic oscillator by Heisenberg’s uncertainty principle. Start by assuming that the product of the uncertainties Δ x and Δ p is at its minimum. Write Δ p in terms of Δ x and assume that for the ground state x Δ x and p Δ p , then write the ground state energy in terms of x . Finally, find the value of x that minimizes the energy and find the minimum of the energy.

proof

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A mass of 0.250 kg oscillates on a spring with the force constant 110 N/m. Calculate the ground energy level and the separation between the adjacent energy levels. Express the results in joules and in electron-volts. Are quantum effects important?

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Questions & Answers

can someone explain normalization condition
Priyojit Reply
1 millimeter is How many metres
Darling Reply
1millimeter =0.001metre
Gitanjali
The photoelectric effect is the emission of electrons when light shines on a material. 
Chris Reply
What is photoelectric effect
Amit Reply
it gives practical evidence of particke nature of light.
Omsai
particle nature
Omsai
photoelectric effect is the phenomenon of emission of electrons from a material(i.e Metal) when it is exposed to sunlight. Emitted electrons are called as photo electrons.
Anil
what are the applications of quantum mechanics to medicine?
Neptune
application of quantum mechanics in medicine: 1) improved disease screening and treatment ; using a relatively new method known as BIO- BARCODE ASSAY we can detect disease-specific clues in our blood using gold nanoparticles. 2) in Genomic medicine 3) in protein folding 4) in radio theraphy(MRI)
Anil
Quantam physics ki basic concepts?
Laxmikanta Reply
why does not electron exits in nucleaus
Kabbo Reply
electrons have negative
YASH
Proton and meltdown has greater mass than electron. So it naturally electron will move around nucleus such as gases surrounded earth
Amalesh
.......proton and neutron....
Amalesh
excuse me yash what negative
Rika
coz, electron contained minus ion
Manish
negative sign rika shrestha ji
YASH
electron is the smallest negetive charge...An anaion i.e., negetive ion contains extra electrons. How ever an atom is neutral so it must contains proton and electron
Amalesh
yes yash ji
Rika
yes friends
Prema
koantam theory
Laxmikanta
yes prema
Rika
quantum theory tells us that both light and matter consists of tiny particles which have wave like propertise associated with them.
Prema
proton and nutron nuclear power is best than proton and electron kulamb force
Laxmikanta
what is de-broglie wave length?
Ramsuphal
plot a graph of MP against tan ( Angle/2) and determine the slope of the graph and find the error in it.
Ime Reply
expression for photon as wave
BARISUA Reply
Are beta particle and eletron are same?
Amalesh Reply
yes
mari
how can you confirm?
Amalesh
sry
Saiaung
If they are same then why they named differently?
Amalesh
because beta particles give the information that the electron is ejected from the nucleus with very high energy
Absar
what is meant by Z in nuclear physic
Shubhu Reply
atomic n.o
Gyanendra
no of atoms present in nucleus
Sanjana
Note on spherical mirrors
Shamanth Reply
what is Draic equation? with explanation
M.D Reply
what is CHEMISTRY
trpathy Reply
it's a subject
Akhter
it's a branch in science which deals with the properties,uses and composition of matter
Eniabire
what is a Higgs Boson please?
FRANKLINE Reply
god particles is know as higgs boson, when two proton are reacted than a particles came out which is used to make a bond between than materials
M.D
bro little abit getting confuse if i am wrong than please clarify me
M.D
the law of refraction of direct current lines at the boundary between two conducting media of
BATTULA Reply

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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