4.1.1 Suppose we adopt a notation
to label the states of a particle in a one-dimensional potential well of thickness L.. Write the bra-ke
notation form that is equivalent to each of the following integrals (do not evaluate the integrals - just
change the notation).
where G is some constant
4.2.1 We will consider the function space that corresponds to all linear functions of a single variable, i.e.,
functions of the form,
defined over the range - -1(i) are orthonormal
Show that the functions and
(ii) By showing that any arbitrary function f(x=ax+b can be represented as the linear
combination show that the functions (1)(x) and y2(x) constitute a
complete basis set for representing such functions.
Represent the function 2x+3 as a vector in a two-dimensional function space by drawing that
vector in a two-dimensional diagram with orthogonal axes corresponding to the functions V1(x)
and y2(x), stating the values of appropriate coefficients or components.
4.6.1 In the notation where functions in a Hilbert space are expressed as vectors in that space, and
operators are expressed as matrices, for functions f) and g) and an operator A, state where each
of the following expressions corresponds to a column vector, a row vector, a matrix, or a complex
4.10.2 Consider the operator
What are the eigenvalues and associated (normalized) eigenvectors of this operator?
(ii) What is the unitary transformation operator that will diagonalize this operator (i.e., th
that will change the representation from the old basis to a new basis in which the operate
represented by a diagonal matrix)? Presume that the eigenvectors in the new basis are
(iii) What is the operator M in this new basis?
4.11.3 Consider the Hermiticity of the following operators.
(i) Prove that the momentum operator is Hermitian. For simplicity you may perform this proof
for a one-dimensional system (i.e., only consider functions of X, and consider only the
[Hints: Consider dx where the V., (x) are a complete orthonormal set.
You may want to consider an integration by parts. Note that the V. (x) must vanish at too,
since otherwise they could not be normalized.]
the operator of Hermitian? Prove your answer.
Is the operator of Hermitian? Prove your
Hints: You may want to consider another integration by parts, and you may presume that the
derivatives dy, (x) also vanish at too.
(iv) Is the operator A = d V (x) Hermitian if V (x) is real? Prove your
4.10.4 Consider the so-called Pauli matrices
(which are used in quantum mechanics as the operators corresponding to the X, y, and 2
components of the spin of an electron, though for the purposes of this problem we can consider them
simply as abstract operators represented by matrices). For this problem, find all the requested
eigenvalues and eigenvectors by hand (i.e., not using a calculator or computer to find the eigenvalues
and eigenvectors), and show your calculations.
(i) Find the eigenvalues and corresponding (normalized) eigenvectors
(ii) Find the eigenvalues and corresponding (normalized) eigenvectors y
of the opera
of the operator o,
(iii) Show by explicit calculation that where i is the identity matrix in this two
(iv) These operators have been represented in a basis that is the set of eigenvectors of à,
Transform all three of the Pauli matrices into a representation that uses the set of eigenvectors of
8, as the basis.
5.1.1 The Pauli spin matrices are quantum mechanical operators that operate in a two-dimensional
Hilbert space, and can be written as
Find the commutation relations between each pair of these operators, proving your answer by explicit
matrix multiplication, and simplifying the answers as much as possible.
5.2.3 Consider the "angular momentum" operators
where p., p. and p2 are the usual momentum operators associated with the X, y, and Z directions.
(Note that these momentum operators are all Hermitian.)
(i) Prove whether on not Lx is Hermitian.
(ii) Construct an uncertainty principle for Ê, and i,
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