Download Quantum Postulates “Mastery of Fundamentals” Questions CH351

Quantum Postulates “Mastery of Fundamentals” Questions
CH351 – Prof. Wu
Here are some questions to test your mastery of the fundamentals of the postulates of
quantum mechanics. Once you’ve mastered the material, you should be able to answer
these questions without reference to your notes or textbook.
1. List the postulates of quantum mechanics and explain them.
Postulate 1: The state of a system is determined by the wavefunction.
Postulate 2: To every classical observable corresponds a linear Hermitian operator in
QM.
Postulate 3: A measurement of the observable with operator Aˆ can only result in an
eigenvalue of that operator.
Postulate 4: Given that the system is in state ψ , the expectation value of the
* ˆ
observable with operator Aˆ is given by A = ∫ ψ€
Aψdx .
all space
Postulate 5: The wave function evolves in time according to the time-dependent
∂Ψ(x,t)€ ˆ
Schroedinger’s €
equation ih
= HΨ(x,t) .
∂t
€
2. What are the properties needed of a wavefunction to be well-behaved?
A wavefunction€is well-behaved if it is normalizable, and if it and its first-derivative
are single-valued, continuous and finite.
3. Why do we say the wavefunction completely specifies the state of a system?
How do we use the wavefunction?
We mean that any physically observable quantity is determined, although perhaps
probabilistically, by the wavefunction alone. The wavefunction is used according to
the postulates above to determine outcomes of measurements.
4. Given a classical observable, write down the corresponding quantum operator.
This is done by writing the classical observable in terms of x and p. Then we
∂
substitute the operators xˆ = x and pˆ = −ih
for x and p. Note the x operator just
∂x
means multiply by x.
€ possible outcomes of observing a quantity, and how is it related to
5. What are the
its quantum operator?€ How do we determine the probabilities associated with
each outcome?
As stated in postulate 3 above, the outcomes can only be the eigenvalues, an, of the
corresponding operator Aˆ , namely Aˆ φ n (x) = an φ n (x) . Given that the system is in a
2
state ψ , the probability of being found in a state j is pn = c n , where the cn are the
€
€
€
€
Quantum Postulates “Mastery of Fundamentals” Questions
CH351 – Prof. Wu
coefficients of the expansion of ψ = ∑ c n φ n (x) in terms of a linear combination of the
eigenfunctions of Aˆ .
n
6. What is an expectation value? How do we determine the expectation value given
€
a wavefunction?
€
The expectation value is the average value of an observable over several
measurements of identical systems with the same wavefunction. It is determined
using postulate 4 above, or by taking an average with a knowledge of the probabilities
of each outcome as given by the answer to the preceding question.
7. What does it mean that two wavefunctions are orthogonal to each other? That a
set of wavefunctions is orthonormal?
Two functions f(x) and g(x) are orthogonal if and only if
∫
f * (x)g(x)dx = 0 , where
all space
x means the positions (one or more dimensional) available to the system or particle.
A set of wavefunctions is orthonormal if and only if any pair of wavefunctions are
orthogonal, each each wavefunction is normalized.
€
8. When do the eigenfunctions of an operator form an orthonormal basis?
The eigenfunctions of a linear Hermitian operator, once normalized, form an
orthonormal basis. Thus the eigenfunctions of any operator corresponding to a
physical observable form an orthonormal basis.
9. What is a commutator? How is the commutator related to whether two quantities
can be observed simultaneously to arbitrary accuracy?
The commutator of operator Aˆ and Bˆ is Aˆ , Bˆ = Aˆ Bˆ − Bˆ Aˆ . Two quantities can be
[ ]
observed simultaneously to arbitrary accuracy if and only if the commutator of their
operators is zero (i.e., if they commute).
€
€
€ the time-dependent and time-independent
10. What is the difference between
Schroedinger’s equation?
The time-dependent Schroedinger’s equation is used to determine how a
wavefunction changes in time. The time-dependent Schroedinger’s equation, which
can be derived from the time-dependent Schroedinger’s equation, is used to find the
stationary states of a system.
Quantum Postulates “Mastery of Fundamentals” Questions
CH351 – Prof. Wu
In addition, you should feel comfortable doing problems like those that have been
assigned in homework. Here are some additional problems you should feel comfortable
doing once you’ve mastered the material.
1. Given a system that consists of two independent degrees of freedom (e.g. two
separated particles, two different masses on springs, one mass moving in the x and
y directions independently), be able to write down the Hamiltonian. From this,
show that the energies are independent and are additive, while the wavefunctions
are products of the independent wavefunctions of each individual system.
Hˆ = Hˆ x + Hˆ y , where Hˆ x is the Hamiltonian for the system in the x direction (PIB,
harmonic oscillator, etc.), and similarly for y. Since Hˆ depends only on x, and
x
€
similarly for y, we can solve the Schroedinger’s equation in each dimension
separately, €
namely Hˆ xψ n x (x) = E n xψ n x (x) and Hˆ y φ n y (y) = E n y φ n y (y) . From this we
can thus show that ψ (x)φ (y) is an eigenfunciton
of Hˆ with eigenvalue
€
nx
ny
y
E nx + E ny :
€
€
ˆ
ˆ
ˆ
Hψ n x (x)φ€n y (y) = H x + H y ψ n x (x)φ n y (y)
(
€
)
€
= Hˆ xψ n x (x)φ n y (y) + Hˆ yψ n x (x)φ n y (y) = φ n y (y) Hˆ xψ n x (x) + ψ n x (x) Hˆ y φ n y (y)
(
)
= φ n y (y)E n xψ n x (x) + ψ n x (x)E n y φ n y (y) = E n x + E n y ψ n x (x)φ n y (y)
€