The Hamiltonian you have provided is a sum of six Pauli matrices, each acting on a different spin. The Kronecker product is a way of multiplying matrices that have the same dimensions. In this case, we can use the Kronecker product to multiply the Pauli matrices together to get a single 64x64 matrix.
import numpy as np
def kron(A, B):
"""
Returns the Kronecker product of A and B.
"""
m, n = A.shape
p, q = B.shape
C = np.zeros((m*p, n*q))
for i in range(m):
for j in range(n):
C[i*p:(i+
1)*p, j*q:(j+1)*q] = A[i, j] * B
return C
H = kron(sigmaz, sigmaz) + kron(sigmaz, sigmaz)
This code first defines a function called kron(), which takes two matrices as input and returns their Kronecker product. The function then creates the Hamiltonian matrix H by multiplying two copies of the Pauli matrix sigmaz together.
Once the Hamiltonian matrix has been created, we can use the eig() function from NumPy to calculate its eigenvalues and eigenvectors. The following code does this:
eigenvalues, eigenvectors = np.linalg.eig(H)
This code first calls the eig() function, which returns the eigenvalues and eigenvectors of H as two separate arrays. The eigenvalues are stored in the eigenvalues array, and the eigenvectors are stored in the eigenvectors array.
The eigenvalues and eigenvectors of the Hamiltonian can be used to understand the energy levels and wavefunctions of the system.
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