Heisenberg Uncertainty Principle ∆𝑥∙∆𝑝≥ℏ2 Or ∆𝐸∙∆𝑡≥ℏ2
Zero Point Energy The origin of zero-point energy is the Heisenberg uncertainty principle. It is the lowest possible energy that a quantum mechanical system may have; it is the energy of its ground state. The most famous such example of zero-point energy is 𝐸=12ℏ𝜔 associated with the ground state of the quantum harmonic oscillator. It is the expectation value of the Hamiltonian of the system in the ground state.
Zero Point Energy 𝐻=ℏ22𝑚 ∇2+𝑉(𝑟,𝑡) 𝑍𝑃𝐸=𝐻=Ψ𝑔𝑟𝑜𝑢𝑛𝑑𝐻Ψ𝑔𝑟𝑜𝑢𝑛𝑑
Vacuum Fluctuations In quantum field theory, the fabric of space is visualized as consisting of fields, with the field at every point in space and time being a quantum harmonic oscillator. The zero-point energy is again the expectation value of the Hamiltonian; here, however, the phrase vacuum expectation value is more commonly used, and the energy is called the vacuum energy. Vacuum energy can also be thought of in terms of virtual particles (also known as vacuum fluctuations) which are created and destroyed out of the vacuum. The concept of vacuum energy was derived from energy-time uncertainty principle.
Vacuum Fluctuations The vacuum state |𝑣𝑎𝑐> of the field is the state of the lowest energy. The expectations values of both 𝑎𝑘𝑠 and 𝑎𝑘𝑠+ vanish in the vacuum state, because: 𝑎𝑘𝑠|𝑣𝑎𝑐> =0=<𝑣𝑎𝑐|𝑎𝑘𝑠+ Vector 𝐹(𝑟,𝑡), which may be the electric or magnetic or the vector potential, having a mode expansion of the general form: 𝐹𝑟,𝑡=1𝐿32𝑘,𝑠𝑙𝜔𝑎𝑘𝑠𝜀𝑘𝑠𝑒𝑖𝑘∙𝑟−𝜔𝑡+h.𝑐
Vacuum Fluctuations Where 𝑙𝜔 is some slowly varying function of frequency which is different for each field vector. Expectation value of 𝐹𝑟,𝑡 in the vacuum state: <𝑣𝑎𝑐𝐹𝑟,𝑡𝑣𝑎𝑐> =0 However, the expectation of the square of the field operator does not vanish, as we will show soon. This implies that there are fluctuations of the em field, even in its lowest energy.
Vacuum Fluctuations If we use the mode expansion and make use of the fact that: 𝑣𝑎𝑐𝑎𝑘𝑠+𝑎𝑘′𝑠′𝑣𝑎𝑐=0 𝑣𝑎𝑐𝑎𝑘𝑠+𝑎𝑘′𝑠′+𝑣𝑎𝑐=0 𝑣𝑎𝑐𝑎𝑘𝑠𝑎𝑘′𝑠′𝑣𝑎𝑐=0 We find that: 𝑣𝑎𝑐𝐹2(𝑟,𝑡)𝑣𝑎𝑐==1𝐿3𝑘𝑠𝑘′𝑠′𝑙𝜔𝑙∗(𝜔′)𝑣𝑎𝑐𝑎𝑘𝑠𝑎𝑘′𝑠′+𝑣𝑎𝑐(𝜀𝑘𝑠∙𝜀𝑘′𝑠′∗)∙𝑒𝑖[𝑘−𝑘′𝑟−𝜔−𝜔′𝑡]
Vacuum Fluctuations 𝑎𝑘𝑠𝑡, 𝑎𝑘′𝑠′+(𝑡)=𝑎𝑘𝑠∙𝑎𝑘′𝑠′+−𝑎𝑘′𝑠′+∙𝑎𝑘𝑠=𝛿𝑘𝑘′3𝛿𝑠𝑠′ With the help of the commutation relation we have: 𝑣𝑎𝑐𝑎𝑘𝑠𝑎𝑘′𝑠′+𝑣𝑎𝑐=𝑣𝑎𝑐(𝑎𝑘′𝑠′+∙𝑎𝑘𝑠+𝛿𝑘𝑘′3𝛿𝑠𝑠′)𝑣𝑎𝑐=𝛿𝑘𝑘′3𝛿𝑠𝑠′
Vacuum Fluctuations So that: 𝑣𝑎𝑐𝐹2(𝑟,𝑡)𝑣𝑎𝑐=1𝐿3𝑘,𝑠𝑙𝜔2=2𝐿3𝑘𝑙𝜔2 ⟶ 22𝜋3𝑙𝜔2𝑑3𝑘 This is clearly non-zero, and indeed is infinite for an unbounded set of modes. As it is know: 𝑣𝑎𝑐∆𝐹2𝑣𝑎𝑐=𝑣𝑎𝑐𝐹2𝑣𝑎𝑐 , ∆𝐹=𝐹−𝐹 ∆𝐹 – the deviation from the mean This shows us that the field fluctuates in the vacuum state.
Vacuum Fluctuations The effects of vacuum energy can be observed in various phenomena such as spontaneous emission, the Casimir effect and the Lamb shift, and are thought to influence the behavior of the Universe on cosmological scales.
Spontaneous Emission Quantum electrodynamics shows that spontaneous emission takes place because there is always some electromagnetic field present in the vicinity of an atom, even when a field is not applied. Like any other system with discretely quantized energy, the electromagnetic field has a zero-point energy. Quantum electrodynamics shows that there will always be some electromagnetic field vibrations present, of whatever frequency is required to induce the charge oscillations that cause the atom to radiate 'spontaneously'.
The Casimir Effect One of the more striking examples is the attractive force between a pair of parallel, uncharged, conducting plates in vacuum. This force is also referred to as a Van der Waals attraction and has been calculated by Dutch physicists Hendrik B. G. Casimir and Dirk Polder (1948).
The Casimir Effect One can account for this force (also known as Casimir force), and obtain an approximate value of its magnitude, by assuming that the force is a consequence of the separation-dependent vacuum field energy trapped between the two plates. If the plats are squares of side L and are separated by a distance z, we may suppose that the system constitutes a “cavity” that supports modes with wave number k down to about 1/z. the vacuum field energy trapped between the plates may therefore be written approximately as: 𝑈=𝑘,𝑠12ℏ𝜔≈𝐿2𝑧1𝑧𝐾ℏ𝑐𝑘 𝑘2𝑑𝑘≈14𝐿2ℏ𝑐𝑧𝐾4−1𝑧3=𝑈𝑢𝑝𝑝𝑒𝑟−𝑈𝑙𝑜𝑤𝑒𝑟
The Casimir Effect we have introduced a high frequency cut-off K to make the energy finite. We can think of the negative rate of change of the lower cut-off energy 𝑈𝑙𝑜𝑤𝑒𝑟 with separation z as constituting a force of attraction, whose magnitude F per unit are is given by: 𝐹=−1𝐿2𝑑𝑈𝑙𝑜𝑤𝑒𝑟𝑑𝑧~ℏ𝑐𝑧4
The Casimir Effect It is interesting to note from the structure of F that the force is proportional to ℏ and is therefore quantum mechanical. Because the strength of the force falls off rapidly with distance, it is only measurable when the distance between the objects is extremely small. On a submicrometre scale, this force becomes so strong that it becomes the dominant force between uncharged conductors.
The Casimir Effect At separations of 10 nm—about 100 times the typical size of an atom—the Casimir effect produces the equivalent of 1 atmosphere of pressure (101.325 kPa), the precise value depending on surface geometry and other factors. In modern theoretical physics, the Casimir effect plays an important role in the chiral bag model of the nucleon; and in applied physics, it is significant in some aspects of emerging micro technologies and nanotechnologies.
The Lamb Shift The Lamb shift, named after Willis Lamb (1913–2008), is a small difference in energy between two energy levels 2S1 / 2 and 2P1 / 2 of the hydrogen atom in quantum electrodynamics. According to Dirac, the 2S1 / 2 and 2P1 / 2 orbitals should have the same energies. However, the interaction between the electron and the vacuum causes a tiny energy shift on 2S1 / 2. Lamb and Robert Retherford measured this shift in 1947. Lamb won the Nobel Prize in Physics in 1955 for his discoveries related to the Lamb shift.
The Lamb Shift In 1948 Welton succeeded in accounting for the Lamb shift between the s and p energy levels of atomic hydrogen in terms of the perturbation of the electronic orbit brought about by vacuum fluctuations. A perturbation 𝛿𝑟 in electronic position in general causes a change of potential energy 𝛿𝑉 given by: 𝛿𝑉=𝑉𝑟+𝛿𝑟−𝑉𝑟=∇𝑉∙𝛿𝑟+12𝜕𝜕𝑟𝑖𝜕𝜕𝑟𝑗𝑉𝛿𝑟𝑖𝛿𝑟𝑗+⋯
The Lamb Shift When we average this over the random displacements 𝑟, the term in 𝛿𝑟2 is the leading non-zero term and we find that: 𝛿𝑉=16∇2𝑉 𝛿𝑟2 In order to calculate the value of 𝛿𝑟2 resulting from the fluctuations of the vacuum field, we observe that, under the influence of an electric field 𝐸𝜔 of frequency 𝜔, the electronic position r obeys the equation of motion: 𝑚𝑟=−𝑒𝐸𝜔cos𝜔𝑡
The Lamb Shift And this results in a mean squared displacement about its equilibrium value of: 𝛿𝑟𝜔2=12𝑒2𝑚2𝜔4 𝐸𝜔2𝑣𝑎𝑐=ℏ𝑒22𝜋3𝜀0𝑚2𝑑3𝑘𝜔3=ℏ𝑒22𝜋2𝜀0𝑚2𝑐3𝜔0Ω𝑑𝜔𝜔
The Lamb Shift The integral diverges logarithmically at the upper end, and had to be provided with a cut-off Ω, which is usually chosen to be of order 𝑚𝑐2/ℏ. When this expression for 𝛿𝑟2 is inserted in 𝛿𝑉, and we average ∇2𝑉(𝑟) over the electronic orbit with the help of the wave function 𝜓(𝑟), we obtain finally for the perturbation of the atomic energy level: 𝛿𝑉=ℏ𝑒212𝜋2𝜀0𝑚2𝑐3 𝑑3𝑟 ∇2𝑉𝑟𝜓𝑟2𝜔0Ω𝑑𝜔𝜔
The Lamb Shift If we take the potential energy 𝑉(𝑟) to be: 𝑉𝑟=−𝑒24𝜋𝜀0𝑟 then: ∇2𝑉𝑟=𝑒2𝜀0𝛿3(𝑟) and the volume integral reduces to: 𝑒2𝜀0𝜓02
The Lamb Shift This vanishes for a p-state but gives a finite value for an s-state. The difference between the s and p energy levels is therefore: ∆𝐸=ℏ𝑒412𝜋2𝜀02𝑚2𝑐3𝜓𝑠02ln𝑚𝑐2ℏ𝜔0
The Lamb Shift This leads to: ∆𝐸ℏ~1040 𝑀𝐻𝑧 For the 2s-state of hydrogen, and is in reasonable agreement with measurements by Lamb and Retherford (1947).
The Beam Splitter After decomposing all fields into plane-wave modes in the usual way, we consider a single incident mode labeled 1, which gives rise to a reflected mode 2 and a transmitted mode 3. r, t are the complex amplitude reflectivity and transmissivity for light incident from one side. 𝑟′, 𝑡′ for light coming from the other side there are no losses in the beam splitter
The Beam Splitter Then these parameters must obey the following reciprocity relations (due to Stokes, 1849): 𝑟=𝑟′ , 𝑡=𝑡′ 𝑟2+𝑡2=1 𝑟𝑡∗+𝑟∗𝑡=0
The Beam Splitter It follows that an incoming classical wave of complex amplitude 𝜐1 gives rise to a reflected wave 𝜐2, and a transmitted wave 𝜐3 such that: 𝑣2=𝑟𝑣1 , 𝑣3=𝑡𝑣1 From these relations it follows immediately that: 𝑣22+𝑣32=𝑡2+𝑟2𝑣12 So that the incoming energy is conserved.
The Beam Splitter Now suppose that we wish to apply a similar argument to the treatment of a quantum field. Then 𝑣1, 𝑣2, 𝑣3 have to be replaced by the complex amplitude operators 𝑎1, 𝑎2 , 𝑎3 , which obey the commutation relations: 𝑎𝑗, 𝑎𝑗+=1, 𝑗=1, 2, 3 𝑎2, 𝑎3+=0
The Beam Splitter if we simply replace 𝑣1, 𝑣2, 𝑣3 by the operators 𝑎1, 𝑎2 , 𝑎3 , we readily find that the commutation equations do not hold for 𝑎2 , 𝑎3. Instead we obtain: 𝑎2 , 𝑎3+=𝑟2𝑎1 , 𝑎1+=𝑟2 𝑎3 , 𝑎3+=𝑡2𝑎1 , 𝑎1+=𝑡2 𝑎2 , 𝑎3+=𝑟𝑡∗𝑎1 , 𝑎1+=𝑟𝑡∗
The Beam Splitter The reason for the discrepancy is that we have ignored the fourth beam splitter input port, which is justifiably ignored in the classical treatment because no light enter that way. However, even if no energy is flowing through the mode labeled 0, in a quantized field treatment there is a vacuum field that enters here and contributes to the two output modes.
The Beam Splitter Accordingly, we need to rewrite the commutation relations: 𝑎2=𝑟𝑎1+𝑡′𝑎0 ; 𝑎3=𝑡𝑎1+𝑟′𝑎0 𝑎2 , 𝑎2+=𝑟2𝑎1 , 𝑎1++𝑡2𝑎0 , 𝑎0+=𝑟2+𝑡2=1 𝑎2 , 𝑎3+=𝑟𝑡∗𝑎1 , 𝑎1+𝑟′∗𝑡′𝑎0 , 𝑎0+=𝑟𝑡∗+𝑟′∗𝑡′=0
Science Fiction or is it? As a scientific concept, the existence of zero point energy is not controversial although the ability to harness it is. Many claims exist of ''over unity devices'' (gadgets yielding a greater output than the required input for operation) driven by zero-point energy. Zero-point energy is not a thermal reservoir, and therefore does not suffer from the thermodynamic injunction against extracting energy from a lower temperature reservoir.
Science Fiction or is it? In 1993 Cole and Puthoff published a thermodynamic analysis, ''Extracting energy and heat from the vacuum'' (see below), in which they concluded that ''extracting energy and heat from electromagnetic zero-point radiation via the use of the Casimir force'' is in principle possible without violating the laws of thermodynamics.
Science Fiction or is it? A thought experiment for a device that readily demonstrates how the Casimir force could be put to use in principle was proposed by physicist Robert Forward in 1984 . A ''vacuum fluctuation battery'' could be constructed consisting of stacked conducting plates. Applying the same polarity charge to all the plates would yield a repulsive force between plates, thereby opposing the Casimir force which is acting to push the plates together. Adjusting the electrostatic force so as to permit the Casimir force to dominate will result in adding energy to the electric field between the plates, thereby converting zero-point energy to electric energy.
Science Fiction or is it? In spite of the dubious nature of these claims (to date no such device has passed a rigorous, objective test), the concept of converting some amount of zero-point energy to usable energy cannot be ruled out in principle.
Conclusion In all the examples above (not the science fiction part of course) we see that the vacuum field plays a fundamental role and is required for internal consistency. The vacuum has certain consequences in quantum electrodynamics that have no counterpart in the classical domain and it cannot be ignored.
Bibliography “Optical Coherence and Quantum Optics” by Leonard Mandel and Emil Wolf “Zero Point Energy and Zero Point Field” – Calphysics Institute Zero Point Energy and Vacuum Energy – Wikipedia General Interest Articles by Matt Visser, Victoria University of Wellington