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| In [[quantum physics]] and [[quantum chemistry]], an '''avoided crossing''' (sometimes called ''intended crossing''<ref>for a less mathematical explanation see [http://goldbook.iupac.org/A00544.html IUPAC Goldbook article]</ref> or ''non-crossing'', ''anticrossing'') is defined as the case when the [[eigenvalue]]s of a [[Hermitian matrix]] representing an observable for a [[Quantum system|system]] and depending on ''N'' continuous real parameters cannot cross (that is, two or more eigenvalues cannot become equal in value) except at a [[manifold]] of ''N''-2 dimensions when the states are symmetric.<ref>Landau,Lifshitz(1981),Quantum Mechanics, p.305</ref> In the case of a [[diatomic molecule]] (one parameter, which describes the [[bond length]]), this means that the eigenvalues do not cross. In the case of a [[diatomic|triatomic]] molecule, this means that the eigenvalues can intersect only at a point (see [[conical intersection]]).
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you have any issues relating to exactly where and how to use [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC], you can make contact with us at the internet site. |
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| This is particularly important in [[quantum chemistry]]. In the [[Born-Oppenheimer approximation]], the [[electronic molecular Hamiltonian]] is [[diagonalizable matrix|diagonalized]] on a set of distinct molecular geometries (the obtained [[eigenvalue]]s are the values of the [[adiabatic]] [[potential energy surface]]s). The geometries for which the potential energy surfaces are avoiding to cross are the [[locus (mathematics)|locus]] where the Born-Oppenheimer approximation fails.
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| ==Avoided crossing in two-state systems==
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| === Emergence of avoided crossing ===
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| Study of a [[two-level system]] is of vital importance in quantum mechanics because it embodies simplification of a lots of physically realizable systems.<ref>https://en.wikipedia.org/wiki/Two-state_quantum_system#Examples_of_two-state_quantum_systems</ref> The effect of [[Perturbation theory (quantum mechanics)|perturbation]] on a two-state system [[Hamiltonian (quantum mechanics)|Hamiltonian]] is manifested through avoided crossings in the plot of individual energy vs energy difference curve of the eigenstates.<ref>Cohen-Tannaoudji,Claude et al.(1992),Quantum Mechanics( Vol. 1), p.409</ref> The two-state Hamiltonian can be written as
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| :<math>\begin{align} H= \begin{pmatrix}E_{1}&0\\0&E_{2}\end{pmatrix} \end{align}\,\!</math> | |
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| The eigenvalues of which are <math>\textstyle E_{1}</math> and <math>\textstyle E_{2}</math> and the [[eigenvector]]s, <math>\textstyle \begin{pmatrix}1\\0\end{pmatrix} </math> and <math>\textstyle \begin{pmatrix}0\\1\end{pmatrix} </math>. These two eigenvectors designate the two states of the system. If the system is prepared in either of the states it would remain in that state. If <math>\textstyle E_{1} </math> happens to be equal to <math>E_{2} </math> there will be a twofold [[Degeneracy (quantum mechanics)|degeneracy]] in the Hamiltonian. In that case any mixed state of the degenerate eigenstates is evidently another eigenstate of the Hamiltonian. Hence the system prepared in any state will remain in that forever. | |
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| [[File:Avoided crossing in two-state system.gif|thumb|320x240px|Avoided crossing in two-state system. The energy level crossing is avoided with increasing the parameter <math>\textstyle w (= |W| )</math>. In the absence of external perturbation the levels would have crossed if the original energy states were degenerate, i.e <math> \textstyle \Delta E = 0 </math>]]
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| However, when subjected to an external [[Perturbation theory (quantum mechanics)|perturbation]], the matrix elements of the Hamiltonian change. For the sake of simplicity we consider a perturbation with only off diagonal elements. Since the overall Hamiltonian must be [[Hermitian matrix|Hermitian]]<ref>http://theory.tifr.res.in/~sgupta/courses/qm2013/hand3.pdf</ref> we may simply write the new Hamiltonian
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| :<math>\begin{align} H^{'} = H + P= \begin{pmatrix}E_{1}&0\\0&E_{2}\end{pmatrix} + \begin{pmatrix}0&W\\W^{*}&0\end{pmatrix} = \begin{pmatrix}E_{1}&W\\W^{*}&E_{2}\end{pmatrix} \end{align}\,\!</math>
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| Where P is the perturbation with zero diagonal terms. The fact that P is Hermitian fixes it's off-diagonal components. The modified eigenstates can be found by diagonalising the modified Hamiltonian. It turns out that the new eigenvalues are,
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| :<math> E_{+}=\frac{1}{2}(E_{1}+E_{2})+\frac{1}{2}\sqrt{(E_{1}-E_{2})^{2}+4|W|^{2}} </math>
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| :<math> E_{-}=\frac{1}{2}(E_{1}+E_{2})-\frac{1}{2}\sqrt{(E_{1}-E_{2})^{2}+4|W|^{2}} </math>
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| If a graph is plotted varying <math>\textstyle (E_{1}-E_{2})</math> as abscissa and <math>\textstyle E_{+}</math> or <math>\textstyle E_{-}</math> as ordinate we find two branches of a hyperbola (as shown in the figure). The curve asymptotically approaches the original unperturbed energy levels. Analyzing the curves it becomes evident that even if the original states were degenerate (i.e <math>\textstyle E_{1}=E_{2} </math> ) the new energy states are no longer equal. However if <math>\textstyle W </math> is set to zero we may find at <math>\textstyle (E_{1}-E_{2})=0 </math>, <math>\textstyle E_{+}=E_{-} </math> and the levels cross. Thus with the effect of the perturbation these level crossings are avoided.
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| === Avoided crossing and quantum resonance ===
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| The immediate impact of avoided level crossing in a degenerate two state system is the emergence of a lowered energy eigenstate. The effective lowering of energy always correspond to increasing stability.<ref>https://en.wikipedia.org/wiki/Energy_minimization</ref> [[Resonance (chemistry)|Bond resonance]] in organic molecules exemplifies the occurrence of such avoided crossings. To describe these cases we may note that the non-diagonal elements in an erstwhile diagonalised Hamiltonian not only modify the energy eigenvalues but also mix the old eigenstates into the new ones.<ref>Cohen-Tannaoudji,Claude et al.(1992),Quantum Mechanics( Vol. 1), p.410</ref> These effects are more prominent if the original Hamiltonian had degeneracy. This mixing of eigenstates to attain more stability is precisely the phenomena of chemical bond resonance.
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| Our earlier treatment started by denoting the eigenvectors <math>\textstyle \begin{pmatrix}1\\0\end{pmatrix} </math> and <math>\textstyle \begin{pmatrix}0\\1\end{pmatrix} </math> as the matrix representation of eigenstates <math>\textstyle |\psi_{1} \rangle </math> and <math>\textstyle |\psi_{2} \rangle </math> of a two-state system. Using [[bra-ket]] notation the matrix elements of <math> H^{'} </math> are actually the terms
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| :<math> H^{'}_{ij}=\langle \psi_{i}|H^{'}|\psi_{j} \rangle </math> with <math> i,j \in \left\{ {1,2}\right\} </math> | |
| where <math> H^{'}_{11}=H^{'}_{22}=E </math> due to the degeneracy of the unperturbed Hamiltonian.
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| The new eigenstates <math>\textstyle |\psi_{+} \rangle </math> and <math>\textstyle |\psi_{-} \rangle </math> can be found by solving the eigenvalue equations <math> H^{'}|\psi_{+}\rangle=E_{+}|\psi_{+}\rangle </math> and <math> H^{'}|\psi_{-}\rangle=E_{-}|\psi_{-}\rangle </math>. From simple calculations it can be shown that
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| :<math> |\psi_{+}\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}e^{i\phi}\\1\end{pmatrix}= \frac{1}{\sqrt{2}} (e^{i\phi}| \psi_{1}\rangle +|\psi_{2}\rangle) </math> and
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| :<math> |\psi_{-}\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}-e^{i\phi}\\1\end{pmatrix}= \frac{1}{\sqrt{2}} (-e^{i\phi}| \psi_{1}\rangle +|\psi_{2}\rangle) </math> where <math> e^{i\phi}=W/|W| </math>
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| It is evident that both of the new eigenstates are mixture of the original degenerate eigenstates and one of the eigenvalues (here <math> E_{-}
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| </math>) is less than the original unperturbed eigenenergy. So the corresponding stable system will naturally mix up the former unperturbed eigenestates to minimize its energy. In the example of [[Benzene]] the experimental evidences of probable bond structures give rise of two different eigenstates, <math>\textstyle |\psi_{1} \rangle </math> and <math>\textstyle |\psi_{2} \rangle </math>. The symmetry of these two structures mandates that <math> \langle \psi_{1}|H|\psi_{1}\rangle=\langle \psi_{2}|H|\psi_{2}\rangle=E </math>.
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| [[File:Benzene delocalization.svg|500px]]
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| However it turns out that the two-state Hamiltonian <math> H </math> of Benzene is not diagonal. The off-diagonal elements result into lowering of energy and the Benzene molecule stabilizes in a structure which is a superposition of these symmetric ones with energy <math> E_{-}<E </math>.<ref>Cohen-Tannaoudji,Claude et al.(1992),Quantum Mechanics( Vol. 1), p.411</ref>
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| For any general two-state system avoided level crossing repels the eigenstates <math>|\psi_{+}\rangle</math> and <math>|\psi_{-}\rangle</math> such that it requires more energy for the system to achieve the higher energy configuration.
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| ==The general avoided crossing theorem==
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| The above illustration of avoided crossing however is a very specific case. From a generalised view point the phenomenon of avoided crossing is actually controlled by the parameters behind the perturbation. For the most general perturbation <math>\textstyle P=\begin{pmatrix}W_{1}&W\\W&W_{2}\end{pmatrix} </math> affecting a two-dimensional [[Linear subspace|subspace]] of the Hamiltonian <math> H </math> we may write the effective Hamiltonian matrix in that subspace as,
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| :<math> \begin{pmatrix}E_{1}&0\\0&E_{2}\end{pmatrix} + \begin{pmatrix}W_{1}&W\\W&W_{2}\end{pmatrix} =\begin{pmatrix}E_{1}+W_{1}&W\\W&E_{2}+W_{2}\end{pmatrix} </math>
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| Here the elements of the state vectors were chosen to be real so that all the matrix elements become real.<ref>Landau,Lifshitz(1981),Quantum Mechanics, p.304</ref>
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| Now the eigenvalues of the system for this subspace is given by
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| :<math> E_{\pm}=\frac{1}{2}(E_{1}+E_{2}+W_{1}+W_{2}) \pm \frac{1}{2}\sqrt{(E_{1}-E_{2}+W_{1}-W_{2})^{2}+4W^{2}} </math>
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| The terms under the square root are squared real numbers. So for these two levels to cross we must simultaneously require
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| :<math> (E_{1}-E_{2}+W_{1}-W_{2})=0 </math>
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| :<math> W=0 </math> | |
| Now if the perturbation <math> P </math> has <math> k </math> parameters <math> { \alpha_{1},\alpha_{2},\alpha_{3}.....\alpha_{k} } </math> we may in general vary these numbers to satisfy these two equations.
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| :<math> (E_{1}-E_{2}+W_{1}-W_{2})=F_{1}(\alpha_{1},\alpha_{2},\alpha_{3}.....\alpha_{k})=0 </math>
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| :<math> W=F_{2}(\alpha_{1},\alpha_{2},\alpha_{3}.....\alpha_{k})=0 </math>
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| If we choose the values of <math> \alpha_{1} </math> to <math> \alpha_{k-1} </math> then both of the equations above has one single free parameter. In general it is not possible to find one <math> \alpha_{k} </math> such that both of the equations are satisfied. However if we allow another parameter to be free both of these two equations will now be controlled by the same two parameters
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| :<math> F_{1}(\alpha_{k-1},\alpha_{k})|_{\alpha_{1},\alpha_{2},...,\alpha_{k-2} \, fixed}=0 </math>
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| :<math> F_{2}(\alpha_{k-1},\alpha_{k})|_{\alpha_{1},\alpha_{2},...,\alpha_{k-2} \, fixed}=0 </math> | |
| And generally there will be two such values of them for which the equations will simultaneously satisfy. So with <math> k </math> distinct parameters <math> k-2 </math> parameters can always be chosen arbitrarily and still we can find two such <math> \alpha_{k} </math>'s such that there would be crossing of energy eigenvalues. In other words the values of <math> E_{+} </math> and <math> E_{-} </math> would be the same for <math> k-2 </math> freely varying [[co-ordinates]](While the rest of the two co-ordinates are fixed from the condition equations). Geometrically the eigenvalue equations describe a [[surface]] in <math> k+1 </math> dimensional space.
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| :<math> E_{\pm}=E_{\pm}(\alpha_{1},\alpha_{2},\alpha_{3}.....\alpha_{k}) </math>
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| Since their intersection is [[Parametrization|parametrized]] by <math> k-2 </math> coordinates we may formally state that for <math> k </math> continuous real parameters controlling the perturbed Hamiltonian, the levels(or surfaces) can only cross at a [[manifold]] of <math> k-2 </math> dimension.<ref>Landau,Lifshitz(1981),Quantum Mechanics, p.305</ref> However the symmetry of the Hamiltonian has a role to play in the dimensionality. If the original Hamiltonian has asymmetric states, <math> \langle \psi_{1}|W|\psi_{2}\rangle \neq \langle \psi_{2}|W|\psi_{1}\rangle </math> ,the off-diagonal terms vanish automatically to ensure hermiticity. This allows us to get rid of the equation <math> W=0 </math>. Now from similar arguments as posed above it is straightforward that for asymmetrical Hamiltonian the intersection of energy surfaces takes place in a manifold of <math> k-1 </math> dimension.<ref>Landau,Lifshitz(1981),Quantum Mechanics, p.305</ref>
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| ==Avoided crossing in polyatomic molecules==
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| In polyatomic molecules there are various parameters which determine the Hamiltonian of the system. The mutual distances between the atoms are definitely one of them. If both of the molecules of a diatomic molecule is same, the symmetry suggests that different configurations keeping their mutual distance fixed will result into same electronic states. So it is the relative distance <math> r </math> which acts as a parameter for the two equations promising level crossing. Hence due to the avoided crossing theorem in general we can not have level crossings between two electronic states of same symmetry.<ref>von Neumann, J. & Wigner, E.P.(1929),Z.Physik 30,467</ref> But in polyatomic molecules the number of independent mutual distances of nuclei are more. For a N-atomic molecule the number of independent mutual separation is <math>\textstyle S=3N-6 </math> (for <math>\textstyle N \ge 2 </math>).Each of them acts as a parameter for the total Hamiltonian. Since we always have minimum of three independent parameters, level crossing is not totally avoided in these molecules.<ref>H. C. Longuet-Higgins,Proc. R. Soc. Lond. A 1975 344,doi: 10.1098/rspa.1975.0095</ref>
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| ==See also== | |
| * [[Adiabatic theorem]]
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| * [[Bond softening]]
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| * [[Bond hardening]]
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| * [[Level repulsion]]
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| * [[Landau–Zener formula]]
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| ==References==
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| {{reflist}}
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum chemistry]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
If you have any issues relating to exactly where and how to use dentist DC, you can make contact with us at the internet site.