ICARUS (experiment): Difference between revisions
en>Bibcode Bot m Adding 0 arxiv eprint(s), 1 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot |
en>Headbomb ce |
||
Line 1: | Line 1: | ||
{{quantum mechanics}} | |||
{{Main|Perturbation theory (quantum mechanics)}} | |||
{{expert-subject|date=March 2012}} | |||
In [[quantum mechanics]], particularly [[Perturbation theory (quantum mechanics)|Perturbation theory]], a '''transition of state''' is a change from an initial [[quantum state]] to a final one. | |||
==Transitions between stationary states== | |||
The following treatment is fairly common in literature<ref>{{cite book|last=C. Harris|first=Daniel|title=Symmetry and spectroscopy|year=1979|publisher=Dover Books|isbn=0-486-66144-X|pages=550|url=http://www.doverpublications.com}}</ref> (though here its slightly adapted), and often referred as time-dependent [[Perturbation theory (quantum mechanics)|perturbation theory]] in a more advanced form. | |||
===Model=== | |||
We assume a one dimensional [[quantum harmonic oscillator]] of [[mass]] ''m'' and [[electric charge|charge]] ''e''. | |||
The expression for the [[potential energy]] of this system is this of the harmonic oscillator. | |||
:<math>V=\dfrac{1}{2}kx^2</math>. | |||
The | The total [[wavefunction]] is denoted by Ψ(''x, t'') (capital [[Psi (letter)|Psi]]), and the spatial part of the wave function is ψ(''x'') (lower case psi). As we deal with [[stationary state]]s, the total wave function is a solution of the [[Schrödinger equation]] and reads | ||
:<math>\Psi(x,t) = \psi(r)\exp\left(-i\dfrac{E}{\hbar}t\right)</math>, | |||
with eigenvalue <math>\textstyle i\hbar\frac{\partial\Psi}{\partial t} = E\Psi</math>. | |||
The probability of transition from the fundamental level labelled 0 to a level labelled 1 under an electromagnetic stimulation is analysed below. | |||
[ | |||
====A two level model==== | |||
For this situation, we write the total wave function as a [[linear combination]] for a two-levels system: | |||
:<math>\Psi(x,t) = c_{0}(t)\Psi_{0}(x,t) + c_{1}(t)\Psi_{1}(x,t)</math> | |||
The coefficients ''c''<sub>0,1</sub> are time-dependent. They represent the proportion of the state (0,1) in the total wave function with time, thus they represent the probability of the wave-function to fall in one of the two state when an ''observer'' | |||
will collapse the wave function. | |||
As we deal with a two-level system, we have the normalisation relation : | |||
:<math>\langle \Psi(x,t) |\Psi(x,t)\rangle =1 \Leftrightarrow \sqrt{|c_{0}(t)|^2+|c_{1}(t)|^2} = 1</math> | |||
====Perturbation==== | |||
The electromagnetic stimulation will be a uniform [[electric field]], oscillating with a [[frequency]] ω. This is very similar to the semi-classical analysis of the behaviour of an [[atom]] or a [[molecule]] under a [[Polarization (waves)|polarized]] [[electromagnetic wave|electromagnetic]] [[plane wave]]. | |||
Thus, potential energy will be the sum of the unperturbed potential and of the perturbation and reads: | |||
:<math>V(x) = \dfrac{1}{2}kx^2+e\epsilon(t)x</math> | |||
===From the Schrödinger equation to ''c''<sub>1</sub> time-dependence=== | |||
The Schrödinger equation will be written : | |||
:<math>\left(-\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2} + | |||
V(x)\right)\Psi(x,t)=i\hbar\dfrac{\partial\Psi(x,t)}{\partial t}</math> | |||
====Energy operator in the Schrödinger equation==== | |||
The time derivative in the right part of the Schrödinger equation reads: | |||
( | :<math>i\hbar\dfrac{\partial\Psi(x,t)}{\partial t} = i\hbar\left(\psi_{0}\exp\left(-i\dfrac{E_{0}t}{\hbar}\right)\left({c_{0}}'(t) -i\dfrac{E_{0}}{\hbar}c_{0}(t)\right) + \psi_{1}\exp\left(-i\dfrac{E_{1}t}{\hbar}\right)\left({c_{1}}'(t) -i\dfrac{E_{1}}{\hbar}c_{1}(t)\right)\right)</math> | ||
:<math>i\hbar\dfrac{\partial\Psi(x,t)}{\partial t} = i\hbar\left(\Psi_{0}\left({c_{0}}'(t) -i\dfrac{E_{0}}{\hbar}c_{0}(t)\right) + \Psi_{1}\left({c_{1}}'(t) -i\dfrac{E_{1}}{\hbar}c_{1}(t)\right)\right)</math> | |||
====Unperturbed hamiltonian==== | |||
On the right part, the total [[hamiltonian (quantum mechanics)|hamiltonian]] is the sum of the unperturbed hamiltonian (without the external electric field) and the external perturbation. This allows to substitute the [[eigenvalues]] of the stationary states in the total hamiltonian. Thus we write: | |||
:<math>\hat{H}\Psi(x,t)=\left(E_{0}c_{0}(t)\Psi_{0}(x,t)+E_{1}c_{1}(t)\Psi_{1}(x,t) + e\epsilon(t)x\Psi(x,t)\right)</math> | |||
Using the Schrödinger equation above, we end up with | |||
:<math> | |||
e\epsilon(t)x\Psi(x,t)=i\hbar(c_{1}'(t)\Psi_{1}(x,t) + c_{0}'(t)\Psi_{0}(x,t)) | |||
</math> | |||
==== Extract the ''c''<sub>1</sub>(''t'') time dependence ==== | |||
We use now the [[bra-ket notation]] to avoid cumbersome integrals. This reads : | |||
:<math> | |||
e\epsilon(t)(c_{1}(t)x|\Psi_{1}(x,t)\rangle + c_{0}(t)x|\Psi_{0}(x,t)\rangle=i\hbar(c_{1}'(t)|\Psi_{1}(x,t)\rangle + c_{0}'(t)x|\Psi_{0}(x,t)\rangle) | |||
</math> | |||
Then we multiply by <math>\langle \Psi_{1} |</math> and end up with the following | |||
:<math> | |||
e\epsilon(t)(c_{1}(t)\langle \Psi_{1} |x|\Psi_{1}\rangle + c_{0}(t)\langle \Psi_{1} |x|\Psi_{0}\rangle)=i\hbar \left(c_{1}'(t)\langle \Psi_{1} |\Psi_{1}\rangle + c_{0}'(t)\langle \Psi_{1} |\Psi_{0}\rangle\right) | |||
</math> | |||
The two different levels are [[orthogonal]], so <math>\langle \Psi_{1}|\Psi_{0}\rangle=0</math>. Also we are working with | |||
normalized wave functions, so <math>\langle \Psi_{1}|\Psi_{1}\rangle=1</math>. | |||
Finally, | |||
:<math> | |||
e\epsilon(t)\left(c_{1}(t)\langle \Psi_{1} |x|\Psi_{1}\rangle + c_{0}(t)\langle \Psi_{1} |x|\Psi_{0}\rangle\right)= | |||
i\hbar c_{1}'(t) | |||
</math> | |||
This latter equation expresses the time variation of ''c''<sub>1</sub> with time. This is the crux of our calculation, | |||
since by then, we can deduce exactly its expression from the differential equation we obtained. | |||
===Solving the time-dependent differential equation=== | |||
There is no proper way in general to evaluate <math>\langle \Psi_{1} |x|\Psi_{0}\rangle</math>, unless we have a precise knowledge of the two unperturbed wave function, that is to say unless we can solve the non-perturbed Schrödinger equation. In the case of the harmonic potential, the wave functions solutions of the one dimensional [[quantum harmonic oscillator]] are known as [[Hermite polynomials]]. | |||
====Establishing the first order differential equation==== | |||
We made several assumptions to get to the final result. First we suppose that c<sub>1</sub>(0) = 0, because at time ''t'' = 0, | |||
the interaction of the field with the matter did not start. That impose for the total wave function to be normalized that | |||
''c''<sub>0</sub>(0) = 1. We use these conditions, and we can write, at ''t'' = 0: | |||
:<math>e\epsilon(t)\langle \Psi_{1} |x|\Psi_{0}\rangle = i\hbar c_{1}'(t)</math> | |||
Again, in this non-relativistic picture, we remove the time dependence outside. | |||
:<math>e\epsilon(t)\exp\left(-i\dfrac{E_{0} - E_{1}}{\hbar}t\right) | |||
\langle \psi_{1} |x|\psi_{0}\rangle = i\hbar c_{1}'(t)</math> | |||
The quantity <math>e\langle \psi_{1} |x|\psi_{0}\rangle </math> is called the [[transition dipole moment|transition moment]] integral. Its [[dimensional analysis|dimensions]] are [charge]·[length] and [[SI units]] A·s·m. | |||
It can be measured experimentally, or calculated analytically if one know the expression of the spatial wave function for both the energy levels. It can be the case if we deal with an harmonic oscillator like it's the case here. We will not it :<math>\mu_{01}</math> as the transition moment from the level 0 to the level 1. | |||
Finally, we end with | |||
:<math>c_{1}'(t) = \dfrac{\mu_{01}}{i\hbar}\epsilon(t)\exp\left(-i\dfrac{E_{0} - E_{1}}{\hbar}t\right)\Rightarrow | |||
c_{1}(t') = \dfrac{\mu_{01}}{i\hbar}\int_{0}^{t'}\epsilon(t)\exp\left(-i\dfrac{E_{0} - E_{1}}{\hbar}t\right)\mathrm{d}t</math> | |||
==== Solving the first order differential equation ==== | |||
The remaining task is to integrate this expression to obtain ''c''<sub>1</sub>(''t''). | |||
However, we must recall from the previous approximations we made, we are here at time ''t'' = 0. | |||
So the solution we obtain from integration will be only valid as long as |''c''<sub>0</sub>(''t'')|<sup>2</sup> is still | |||
very close to 1, that is to say for very short time after the perturbation began to act. | |||
We suppose that the time dependent perturbation has the following form, to make | |||
the computation easier. | |||
:<math>\epsilon(t)=\epsilon_{0}\exp(i\omega t)</math> | |||
This is a scalar quantity, as we assumed from the beginning a scalar charged | |||
particle and a one | |||
dimensional electric field. | |||
So we have to integrate the following expression : | |||
:<math>c_{1}(t') = | |||
\dfrac{\mu_{01}\epsilon_0}{i\hbar}\int_{0}^{t'}\mathrm{d}t\exp\left(-i\left(\dfrac{E_{0} | |||
- E_{1}}{\hbar} - | |||
\omega\right)t\right)</math> | |||
We can write | |||
:<math>c_{1}(t') = \dfrac{\mu_{01}\epsilon_0}{i\hbar} \int_{0}^{t'} \mathrm{d}t \exp\left(-i\frac{E_{0}- E_{1}}{\hbar}t\right) | |||
\exp({i\omega t})=\int_{-\infty}^{+\infty} \mathrm{d}t \exp\left(-i\frac{E_{0}-E_{1}}{\hbar} t\right)H\left(\frac{t}{t'}-\frac{1}{2}\right) \exp\left(i\omega t\right)</math> | |||
and doing the variable change <math>t\rightarrow -t</math> we obtain the correct form of the Fourier transform : | |||
:<math>c_{1}(t') = \int_{-\infty}^{+\infty} \mathrm{d}t \exp\left(i\frac{E_{0}-E_{1}}{\hbar} t\right)H\left(-\frac{t}{t'}-\frac{1}{2}\right) \exp\left(-i2\pi\nu t\right)</math> | |||
==== Using the Fourier transform ==== | |||
where <math>H</math> is the [[rectangular function]]. We notice from the previous equation that ''c''<sub>1</sub>(''t'') is the [[Fourier transform]] of the product of a cosine with a square of width ''t'''. From then, the formalism of Fourier transforms will make the work easier. | |||
We have | |||
:<math>c_{1}(t') = \mathrm{TF}\left[\exp{\left(i\frac{E_{0} | |||
- E_{1}}{\hbar} t\right)}H\left(-\frac{t}{t'}-\dfrac{1}{2}\right)\right] = \mathrm{TF}\left[\exp{\left(i\frac{E_{0} | |||
- E_{1}}{\hbar} t\right)}\right]\otimes\mathrm{TF}\left[H\left(-\frac{t}{t'}-\dfrac{1}{2}\right)\right]</math> | |||
:<math>c_{1}(t') = \delta\left(f-\dfrac{E_{0}-E_{1}}{h}\right)\otimes\mathrm{TF}\left[H\left(\frac{t}{t'}-\dfrac{1}{2}\right)\right]</math> | |||
:<math>c_{1}(t') = \delta\left(f-\dfrac{E_{0}-E_{1}}{h}\right)\otimes(\exp({i\pi f t'})\mathrm{sinc}(t'f))</math> | |||
Where sinc is the [[Sinc_function|cardinal sinus]] function in its normalized form. The convolution with the [[Dirac distribution]] will translate the term on the left of the <math>\otimes</math> sign. | |||
We obtain finally | |||
:<math>c_1(t') = \exp\left({i\pi \left(f-\dfrac{E_{0}-E_{1}}{h}\right) t'}\right) \mathrm{sinc}\left(t'\left(f-\dfrac{E_{0}-E_{1}}{h}\right)\right)</math> | |||
=== Interpretation === | |||
The probability of a transition is given in general for a multi-level system by the following expression:<ref>Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition), R. Eisberg, R. Resnick, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0</ref> | |||
:<math>P_k=\sum_{n \neq k}c_n^*(t)c_n(t)</math> | |||
==== Final result ==== | |||
The probability to fall in the ''1'' state corresponds to <math>|c_{1}(t)|^2</math>. This is really easy to compute from all the tedious calculation we made previously. We observe in the equation that <math>|c_{1}(t)|^2</math> has a very simple expression. Indeed, the phase factor, varying with ''t'', disappears naturally. | |||
So we obtain the expression | |||
:<math>|c_{1}(t)|^2=\mathrm{sinc}^2\left(t\left(f-\dfrac{E_{0}-E_{1}}{h}\right)\right)</math> | |||
=== Conclusion === | |||
We made the hypothesis that the stimulation was a complex exponential. However a true electric field is real valued. A further analysis should take it in account. Also, we always assume that ''t'' is very small. We should keep it in mind before to conclude. | |||
== References == | |||
{{Reflist}} | |||
==Further reading== | |||
* ''Quantum mechanics'', E. Zaarur, Y. Peleg, R. Pnini, Schaum’s Oulines, Mc Graw Hill (USA), 1998, ISBN (10-) 007-0540187 | |||
* ''Quantum mechanics'', E. Zaarur, Y. Peleg, R. Pnini, Schaum’s Easy Oulines Crash Course, Mc Graw Hill (USA), 2006, ISBN (10-)007-145533-7 ISBN (13-)978-007-145533-6 | |||
* ''Quantum Mechanics Demystified'', D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9 | |||
* ''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0 | |||
* ''Stationary States'', A. Holden, College Physics Monographs (USA), Oxford University Press, 1971, ISBN 0-19-851121-3 | |||
==See also== | |||
*[[Quantum number]] | |||
*[[Vacuum#The quantum-mechanical vacuum|Quantum mechanic vacuum]] or [[vacuum state]] | |||
*[[Virtual particle]] | |||
*[[Steady State]] | |||
*[[Operator (physics)]] | |||
*[[Probability]] | |||
*[[Integration (mathematics)|Integration]] | |||
*[[Differential equation]] | |||
*[[Numerical analysis]] | |||
[[Category:Quantum mechanics]] |
Revision as of 23:29, 15 September 2013
In case you look at the checklist of the Forbes 40 richest folks in Singapore, you will notice many who made their fortunes growing and investing in real estate. Or simply look around you – the average Singaporean's wealth in all probability comes more from the appreciation of his HDB flat or non-public property than from another asset.
As my husband and I are also Singapore PR, we plan to take our criticism to our MP as properly, hopefully the government will turn into conscious of this drawback or perhaps they knew about it already, but it is time to take some action to right this lengthy standing grievance among patrons, and even amongst brokers who acquired shunned out from offers like my poor agent. She misplaced a sale for certain and unbeknownst to her, she in all probability never had an opportunity with that property proper from the start! I feel sorry for her, and appreciated her effort in alerting us about this unit and attempting to get us the property; but I'm a bit of aggravated or perhaps a bit resentful that we lost that condo basically as a result of we had been attached to her or any co-broke agent?
A very powerful part of the process of finding and securing housing is finding a superb estate agent, which is greatest achieved by word of mouth. As soon as expats have managed this feat, what follows is significantly less complicated and less annoying. Steps to renting property in Singapore We've got collected a cross-section of the most effective property resources and repair providers, multi function simple-to-use location, to additional streamline your property purchase or sale. Whether or not you are a first time home buyer or a seasoned seller, you will discover SingaporePropertyExchange.com the essential resource to strengthen your property transaction. iii) for such different period as the Registrar might enable, and had been resident in Singapore throughout that period. ECG Property Pte. Ltd.
Have passed an industry examination i.e Common Examination for House Agents (CEHA) or Actual Estate Company (REA) exam, or equal; Exclusive agents are extra willing to share listing data thus guaranteeing the widest doable coverage inside the actual property group thru Multiple Listings and Networking. Accepting a critical provide is easier since your agent is fully aware of all marketing exercise related together with your property. This reduces your having to verify with a number of agents for another affords. Price control is well achieved. Paint work in good restore-discuss along with your Property Advisor if main works are nonetheless to be accomplished. Softening in residential property costs continue, led by 2.eight per cent decline in the index for Remainder of Central Area
With the steam of an overheated property market dying down, the excitement has now shifted to builders playing discount video games to push new ec launch singapore [that guy] initiatives or clear outdated stock. With so many ‘great offers', patrons are spoiled for choices. In case you overprice your house, buyers shall be delay by the worth. Because of this your own home may take longer to sell, and end up being sold at a lower cost. Patrons may think that you are desperate in promoting (so that you have interaction multiple brokers to sell it off quick). Since they think you might be determined, they may not offer you a great value. Additionally, estate agents are responsibility-certain to avoid another potential conflicts of interest (equivalent to if a celebration to the transaction is related to the agent) until the client's waiver is obtained.
Like anywhere else, cash goes farther when you're exterior the prime districts. In case you plan to stay in a central location, you will have to settle for much less space or fewer onsite services. While rental prices are nonetheless high, 2013 has seen costs adjusting downwards in the higher brackets above $10,000 and the central districts flat-lining. Nevertheless, good value properties in the outer regions and decrease priced items have retained and even seen will increase as individuals typically have been downsizing
It's also usually free to tenants. However tenants paying lower than $three,500 monthly in rental typically pay a half month commission for annually of rental. For rentals above $3,500, tenants don't pay any commission. Does the acquisition process differ for residential, retail, office and industrial properties? Our Guide will assist you to have a head begin within the SEARCH ENGINE OPTIMIZATION bandwagon by performing the right WEBSITE POSITIONING implementation with our custom-made WEBSITE POSITIONING ranking tatics. Basic Search Engine Optimization Capacity to add / edit or delete Challenge Change Ordering of Venture Unpublish or Publish a Undertaking Capability to add / edit or delete a banner of the Undertaking I feel what amazes us firstly is that each owner and agent have to be there with us! Month-to-month rent is $3,888
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
In quantum mechanics, particularly Perturbation theory, a transition of state is a change from an initial quantum state to a final one.
Transitions between stationary states
The following treatment is fairly common in literature[1] (though here its slightly adapted), and often referred as time-dependent perturbation theory in a more advanced form.
Model
We assume a one dimensional quantum harmonic oscillator of mass m and charge e. The expression for the potential energy of this system is this of the harmonic oscillator.
The total wavefunction is denoted by Ψ(x, t) (capital Psi), and the spatial part of the wave function is ψ(x) (lower case psi). As we deal with stationary states, the total wave function is a solution of the Schrödinger equation and reads
The probability of transition from the fundamental level labelled 0 to a level labelled 1 under an electromagnetic stimulation is analysed below.
A two level model
For this situation, we write the total wave function as a linear combination for a two-levels system:
The coefficients c0,1 are time-dependent. They represent the proportion of the state (0,1) in the total wave function with time, thus they represent the probability of the wave-function to fall in one of the two state when an observer will collapse the wave function.
As we deal with a two-level system, we have the normalisation relation :
Perturbation
The electromagnetic stimulation will be a uniform electric field, oscillating with a frequency ω. This is very similar to the semi-classical analysis of the behaviour of an atom or a molecule under a polarized electromagnetic plane wave.
Thus, potential energy will be the sum of the unperturbed potential and of the perturbation and reads:
From the Schrödinger equation to c1 time-dependence
The Schrödinger equation will be written :
Energy operator in the Schrödinger equation
The time derivative in the right part of the Schrödinger equation reads:
Unperturbed hamiltonian
On the right part, the total hamiltonian is the sum of the unperturbed hamiltonian (without the external electric field) and the external perturbation. This allows to substitute the eigenvalues of the stationary states in the total hamiltonian. Thus we write:
Using the Schrödinger equation above, we end up with
Extract the c1(t) time dependence
We use now the bra-ket notation to avoid cumbersome integrals. This reads :
Then we multiply by and end up with the following
The two different levels are orthogonal, so . Also we are working with normalized wave functions, so .
Finally,
This latter equation expresses the time variation of c1 with time. This is the crux of our calculation, since by then, we can deduce exactly its expression from the differential equation we obtained.
Solving the time-dependent differential equation
There is no proper way in general to evaluate , unless we have a precise knowledge of the two unperturbed wave function, that is to say unless we can solve the non-perturbed Schrödinger equation. In the case of the harmonic potential, the wave functions solutions of the one dimensional quantum harmonic oscillator are known as Hermite polynomials.
Establishing the first order differential equation
We made several assumptions to get to the final result. First we suppose that c1(0) = 0, because at time t = 0, the interaction of the field with the matter did not start. That impose for the total wave function to be normalized that c0(0) = 1. We use these conditions, and we can write, at t = 0:
Again, in this non-relativistic picture, we remove the time dependence outside.
The quantity is called the transition moment integral. Its dimensions are [charge]·[length] and SI units A·s·m.
It can be measured experimentally, or calculated analytically if one know the expression of the spatial wave function for both the energy levels. It can be the case if we deal with an harmonic oscillator like it's the case here. We will not it : as the transition moment from the level 0 to the level 1.
Finally, we end with
Solving the first order differential equation
The remaining task is to integrate this expression to obtain c1(t). However, we must recall from the previous approximations we made, we are here at time t = 0. So the solution we obtain from integration will be only valid as long as |c0(t)|2 is still very close to 1, that is to say for very short time after the perturbation began to act.
We suppose that the time dependent perturbation has the following form, to make
the computation easier.
This is a scalar quantity, as we assumed from the beginning a scalar charged particle and a one dimensional electric field.
So we have to integrate the following expression :
We can write
and doing the variable change we obtain the correct form of the Fourier transform :
Using the Fourier transform
where is the rectangular function. We notice from the previous equation that c1(t) is the Fourier transform of the product of a cosine with a square of width t'. From then, the formalism of Fourier transforms will make the work easier.
We have
Where sinc is the cardinal sinus function in its normalized form. The convolution with the Dirac distribution will translate the term on the left of the sign.
We obtain finally
Interpretation
The probability of a transition is given in general for a multi-level system by the following expression:[2]
Final result
The probability to fall in the 1 state corresponds to . This is really easy to compute from all the tedious calculation we made previously. We observe in the equation that has a very simple expression. Indeed, the phase factor, varying with t, disappears naturally.
So we obtain the expression
Conclusion
We made the hypothesis that the stimulation was a complex exponential. However a true electric field is real valued. A further analysis should take it in account. Also, we always assume that t is very small. We should keep it in mind before to conclude.
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
Further reading
- Quantum mechanics, E. Zaarur, Y. Peleg, R. Pnini, Schaum’s Oulines, Mc Graw Hill (USA), 1998, ISBN (10-) 007-0540187
- Quantum mechanics, E. Zaarur, Y. Peleg, R. Pnini, Schaum’s Easy Oulines Crash Course, Mc Graw Hill (USA), 2006, ISBN (10-)007-145533-7 ISBN (13-)978-007-145533-6
- Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9
- Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
- Stationary States, A. Holden, College Physics Monographs (USA), Oxford University Press, 1971, ISBN 0-19-851121-3
See also
- Quantum number
- Quantum mechanic vacuum or vacuum state
- Virtual particle
- Steady State
- Operator (physics)
- Probability
- Integration
- Differential equation
- Numerical analysis
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition), R. Eisberg, R. Resnick, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0