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visual basic print barcode label This is the wave equation for relativistic strings using lightcone coordinates. in Java
This is the wave equation for relativistic strings using lightcone coordinates. Scan QRCode In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Printing QR In Java Using Barcode creator for Java Control to generate, create QR Code image in Java applications. Solutions of the Wave Equation
QR Code ISO/IEC18004 Reader In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Barcode Encoder In Java Using Barcode drawer for Java Control to generate, create barcode image in Java applications. In the next chapter, we will consider the hamiltonian and stressenergy tensor and write down conserved charges and currents for the string. Right now, let s focus on nding a solution of the wave equation given in Eq. (2.46). From elementary mechanics, we know that the solution of a wave equation can be written in terms of a superposition of waves moving to the left on the string and waves moving to the right on the string. If the motion is in one dimension (call it x), then we can write down a solution of the form f (t , x ) = fL ( x vt ) + fR ( x + vt ) We will write the equations of motion for the relativistic string in the same way. We have a solution which is a superposition of leftmoving components X L ( + ) and rightmoving components X R ( ): Decoding Bar Code In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Creating QR Code 2d Barcode In C#.NET Using Barcode maker for VS .NET Control to generate, create Denso QR Bar Code image in VS .NET applications. X ( , ) = X L ( + ) + X R ( ) Printing QR In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. QR Code ISO/IEC18004 Drawer In .NET Framework Using Barcode creator for .NET framework Control to generate, create QR Code 2d barcode image in .NET framework applications. (2.47) Generate QRCode In Visual Basic .NET Using Barcode printer for Visual Studio .NET Control to generate, create Quick Response Code image in VS .NET applications. Generate Code 128 Code Set C In Java Using Barcode generation for Java Control to generate, create Code 128 Code Set C image in Java applications. You should recall from partial differential equations that the most general solution can be written as an expansion of Fourier modes. Here, we denote these modes as k , and write the leftmoving and rightmoving components as Printing UCC128 In Java Using Barcode encoder for Java Control to generate, create UCC128 image in Java applications. Bar Code Creator In Java Using Barcode printer for Java Control to generate, create bar code image in Java applications. 2 k ik ( + ) x s s X ( , ) = + p ( + ) + i e 2 2 2 k 0 k
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Draw 1D In .NET Using Barcode creation for .NET Control to generate, create 1D image in .NET applications. Bar Code Generator In Java Using Barcode maker for BIRT Control to generate, create barcode image in Eclipse BIRT applications. (2.49) String Theory Demysti ed
We have introduced some new terms here. First, we have included the characteristic length of the string which is related to the Regge slope parameter and hence to the tension in the string via T= 1 2 1 2 = (2.50) Next, notice the coordinate x and momentum p . These are the center of mass coordinate and the total momentum of the string, respectively. The zeroth order Fourier mode is de ned in terms of 0 = 0 = (2.51) What does this tell us physically The solutions imply that the string can move as a single unit with position and momentum through spacetime. In addition, it also has vibrations, which are described by the modes k . When you see modes like this, you should think quantization (think in terms of the harmonic oscillator or elds in quantum eld theory). Remember, we are still in the realm of classical physics, even if it s relativistic classical physics. So the solutions of the wave equation X , X L , and X R must be real functions. This implies that x and p are real (as they must be, given their physical interpretation) and allows us to relate positive and negative modes k = k
k = k
(2.52) where * represents the complex conjugate. Now, let s take a look at the solutions of the wave equation with different boundary conditions. Open Strings with Free Endpoints
Open strings with free endpoints satisfy the Neumann boundary condition that we reproduce here: X =0 when = 0, CHAPTER 2 Equations of Motion
Now, looking at Eqs. (2.48) and (2.49) we see that
2 X L = s p + 2 2 X R = s p 2 s
k 0 s
ik ( + ) k
2 k 0 ik ( + ) k
Summing these as in Eq. (2.47) and setting = 0, X =0 0 =
( p p ) +
( 2 s k 0
k e ik
This tells us that in the case of an open string with free endpoints, it must be the case that p = p (string cannot wind around itself ) (same modes for left and rightmoving waves) = Physically, this means that for an open string with free endpoints the modes combine to form standing waves on the string. Before we move on to our next case, let s consider the other boundary condition, which is imposed at the other end of the string = . 0= = = X L
= X R
= 2 k 0 ik ( + ) k
p 2 s k 0
ik ( + ) k
2 k 0 s
ik k
e ik eik ( 2i ) 2i sin( k ) =i 2 k 0 ik k
String Theory Demysti ed
This can only be true if sin k = 0 , which means that k must be an integer. Denoting it by n, a simple exercise shows that Eq. (2.47) can be written as X = x +

