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From Quantum Mechanics to Lagrangian Densities Just as there is no derivation of quantum mechanics from classical mechanics, there is no derivation of.

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Presentation on theme: "From Quantum Mechanics to Lagrangian Densities Just as there is no derivation of quantum mechanics from classical mechanics, there is no derivation of."— Presentation transcript:

1 From Quantum Mechanics to Lagrangian Densities Just as there is no derivation of quantum mechanics from classical mechanics, there is no derivation of relativistic field theory from quantum mechanics. The route from one to the other is based on physically reasonable postulates and the imposition of Lorentz invariance and relativistic kinematics. The final theory is a model whose survival depends absolutely on its success in producing numbers which agree with experiment.

2 Summary: Quantum Mechanics Momentum Becomes an operator The ten minute course in QM. and use The Hamiltonian becomes an operator.

3 This condition places a strong mathematical condition on the wave function. Physical interpretation of the wave function.

4 Note that the Schrodinger equation reflects this relationship

5 Quantization arises from placing boundary conditions on the wave function. It is a mathematical result! L = r x P = r x

6 A toy model postulate approach to quantum field theory

7 p- p p -

8 Note that * (r,t) (r,t) does not represent the probability per unit volume density of the particle being at (r,t).

9 --

10 The resulting wave equation:

11 The negative energy states arose from This emerges from starting out with We know the energy of a real particle cant be < 0..

12 Suppose the mass, m, is zero: This is the same equation we derived from Maxwells equations for the A vector (except, of course, above we have a scalar, ).

13 Following the same derivation used for the A vector we have : We try a solution of the form Solving the wave equation This is just like the E & M equation – except for the mass term.

14 We can see that after taking the partial derivatives there is a condition on the components of k and k 0. The k and k 0 (with p= k and p 0 = k 0 ) must satisfy the same conditions as a relativistic particle with rest mass, m. Note, now the 4-dimensional dot product cannot = 0.

15 We have the following two linearly independent solutions to the wave equation: The most general (complete) solution to the wave equation is

16 The field operator for a neutral, spin =0, particle is creates a single particle with momentum p= k and p 0 = k 0 at (r,t) Destroys a single particle with momentum p= k and p 0 = k 0 at (r,t)

17 In quantum field theory, the Euler-Lagrange equations give the particle wave function! Lagrangians and the Lagrangian Density Recall that, and the Euler-Lagrange equations give F = ma

18 This calls for a different kind of Lagrangian -- not like the one used in classical or quantum mechanics. So, we have another postulate, defining what is meant by a Lagrangian – called a Lagrangian density. d/dt in the classical theory

19 Note that the Lagrangian density is quadratic in (r,t) and the Lorentz invariance is satisfied by using µ and µ

20 We can apply the Euler Lagrange equations to the above L: This part is easy. This part has a very simple result but it is hard to carry out.

21 Finally, give the wave equation for the neutral spin = 0 particle. The Euler-Lagrange Equations with this Lagrangian density

22 Summary for neutral (Q=0) scalar (spin = 0) particle,, with mass, m. Lagrangian density wave equation field operator

23

24 Charged (q = ±e) scalar (spin =0) particle with mass, m

25 8 terms cancel charged scalar particle

26 and the Euler-Lagrange equation From the Lagrangian density we can derive the wave equation

27 creates positively charged particle with momentum p= k and p 0 = k 0 at (r,t) destroys negatively charged particle with momentum p= k and p 0 = k 0 at (r,t) creates negatively charged particle with momentum p= k and p 0 = k 0 at (r,t) destroys negatively charged particle with momentum p= k and p 0 = k 0 at (r,t)

28 The scalar field (which represents a boson) must also satisfy a special boson commutation property: Creation and annihilation operators with the same k dont commute. Everything else commutes.

29 Example of how the commutation relation is used We will use this when we calculate the charge of a particle. Later we will find that fermions satisfy a different commutation relation.

30 What follows is for the graduate students.

31 For the graduate students:

32 For the undergraduates: You can just remember this

33 Before taking the partial derivative, it is helpful to rewrite the L. Let = in this summation, so it does not become confused with the index of / x -- used later.

34 For the undergraduates: You can just remember this


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