Computational Physics

2.2 A program for calculating cross sections 23 l= 6 l= 5 l= 4 l= 0 r[
] –4 –2 0 2 4 0 0.5 1 1.5 2 2.5 Veff (r)[meV] –6 6 Figure ...

24 Quantum scattering with a spherically symmetric potential 10 15 20 25 30 35 40 45 0 0.5 1 1.5 2 2.5 3 3.5 l= 4 l= 5 l= 6 Ener ...

2.3 Calculation of scattering cross sections 25 *2.3 Calculation of scattering cross sections In this section we derive Eqs. (2. ...

26 Quantum scattering with a spherically symmetric potential f(θ )can also be written as an expansion in Legendre polynomials: f ...

Exercises 27 Exercises 2.1 [C] Try using the Runge–Kutta method with an adaptive time step to integrate the radial Schrödinger e ...

28 Quantum scattering with a spherically symmetric potential Lennard–Jones potential without cut-off is given by the phase shift ...

3 The variational method for the Schrödinger equation 3.1 Variational calculus Quantum systems are governed by the Schrödinger e ...

30 The variational method for the Schrödinger equation change inEvanishes to first order: δE≡0. (3.2) Defining P=〈ψ|H|ψ〉 and Q=〈 ...

3.1 Variational calculus 31 (5) (5) (5) (5) 1 1 1 2 2 2 3 3 3 4 (^44) (4) (4) (4) (4) E E E E E E E E E E E E Figure 3.1. The be ...

32 The variational method for the Schrödinger equation functions. Because the computer time needed for matrix diagonalisation sc ...

3.2 Examples of variational calculations 33 numerical library; these can be found in the NETLIB repository). In the following we ...

34 The variational method for the Schrödinger equation Table 3.1.Energy levels of the infinitely deep potential well. N= 5 N= 8 ...

3.2 Examples of variational calculations 35 where the second term in the square brackets is the Coulomb attraction potential of ...

36 The variational method for the Schrödinger equation If the program works correctly, it should yield a value close to the exac ...

3.4 Perturbation theory and variational calculus 37 Then we can rewrite(3.13)as V†HVV−^1 C=EV†SVV−^1 C (3.31) and, defining C′=V ...

38 The variational method for the Schrödinger equation shall use the following notation:nandmlabel the states inA,αandβlabel the ...

Exercises 39 so UC=EC. (3.46) This equation is similar to(3.11), except thatHis replaced byU. Notice that Udepends on the energy ...

40 The variational method for the Schrödinger equation (Note that, in general, the lowest-but-one variational eigenstate 1 is n ...

References 41 It is important to note that, apart from the approximation involved in having a finite basis set, there is another ...

42 The variational method for the Schrödinger equation [2] J. M. Thijssen and J. E. Inglesfield, ‘Embedding muffin tins into a f ...