Let {|a_{n}>} be an orthonormal basis
of eigenvectors of the operator A, A|a_{n}> = a_{n}|a_{n}>.
Assume that the eigenvalues are not degenerate. Let |Ψ_{1}> and
|Ψ_{2}>
be two normalized eigenvectors of the operator B with eigenvalues b_{1}
and b_{2} respectively.

B|Ψ_{1}> = b_{1}|Ψ_{1}>, B|Ψ_{2}> =
b_{2}|Ψ_{2}>.

(If B is the Hamiltonian H, then b_{1} = E_{1}
and b_{2} = E_{2}.)

Assume A and
B do not commute.

Assume at t = 0 the system is in the state |Ψ_{1}>.
(If B is the Hamiltonian, then the system is in a stationary
state.)

The probability that a measurement of A will yield the eigenvalue a_{n}
is P_{1}(a_{n}) = |<a_{n}|Ψ_{1}>|^{2}. (4^{th}
postulate of quantum mechanics)

Similarly, if at t = 0 the
system is in state |Ψ_{2}> then P_{2}(a_{n}) = |<a_{n}|Ψ_{2}>|^{2}.

Now consider a normalized state |Ψ> which is a **linear superposition**
of |Ψ_{1}> and |Ψ_{2}>.

The system is not in an eigenstate of B.

(If B is the
Hamiltonian, then the system is not in a stationary state, it is in a **coherent** state.)

|Ψ> = λ_{1}|Ψ_{1}> + λ_{1}|Ψ_{2}>,
<Ψ|Ψ> = 1 --> |λ_{1}|^{2} + |λ_{2}|^{2}
= 1.

The probability that a measurement of B will yield b_{1} is |<Ψ_{1}|Ψ>|^{2}
= |λ_{1}|^{2}.

The probability that a measurement of B
will yield b_{2} is |<Ψ_{2}|Ψ>|^{2} = |λ_{2}|^{2}.

The
probability that a measurement of A will yield a_{n} is

P(a_{n}) =|<a_{n}|Ψ>|^{2}
= <a_{n}|Ψ><Ψ|a_{n}>

= (λ_{1}<a_{n}|Ψ_{1}> + λ_{2}<a_{n}|Ψ_{2}>)(
λ_{1}*<Ψ_{1}|a_{n}> + λ_{2}*<Ψ_{2}|a_{n}>)

= |λ_{1}|^{2}|<a_{n}|Ψ_{1}>|^{2}
+| λ_{2}|^{2}|<a_{n}|Ψ_{2}>|^{2} +
λ_{1}*λ_{2}<a_{n}|Ψ_{2}><Ψ_{1}|a_{n}>
+ λ_{1}λ_{2}*<a_{n}|Ψ_{1}><Ψ_{2}|a_{2}>

= |λ_{1}|^{2}P_{1}(a_{n}) + |λ_{2}|^{2}P_{2}(a_{n})
+ 2Re{ λ_{1}λ_{2}*<a_{n}|Ψ_{1}><Ψ_{2}|a_{2}>}

≠ |λ_{1}|^{2}P_{1}(a_{n}) + |λ_{2}|^{2}P_{2}(a_{n}).

If we had a **statistical mixture** of states |Ψ_{1}> and |Ψ_{2}>
with weights |λ_{1}|^{2} and |λ_{2}|^{2} respectively, i.e. if we
had a collection of systems with some in state |Ψ_{1}>
and some in state |Ψ_{2}>, then the
probability of measuring b_{i} (i = 1, 2) would be |λ_{i}|^{2} and the probability of measuring a_{n }would
be λ_{1}|^{2}P_{1}(a_{n}) + |λ_{2}|^{2}P_{2}(a_{n}).

**
A linear superposition is not a statistical mixture**.

If a system is in a linear superposition of eigenstates of an observable B and
we measure an observable A which does not commute with B, then we must
take interference effects into account when
predicting the result of the measurement.

**Quantum beats
**Let B = H, b

Then U(t,0) = exp(-iHt/ħ), |Ψ

|Ψ(0)> = λ

|Ψ(t)> = U(t,0)|Ψ(0)> = λ

Now the cross term in P(a

i.e. it becomes time dependent and oscillates with a frequency f

P(a

We may consider P(a_{n}) = |<a_{n}|Ψ>|^{2}
as the square of the probability
amplitude. The probability amplitude <a_{n}|Ψ> = λ_{ 1}<a_{n}|Ψ_{1}>
+ λ_{2}<a_{n}|Ψ_{2}> is the weighted sum of the probability amplitudes P_{1}(a_{n})
and P_{2}(a_{n}) . To obtain the probability P(a_{n}) for
a linear superposition of states, we take the square of the weighted sum of the
probability amplitudes, **not** the sum of the squares.

(1) Without giving the system time to evolve, measure C. The probability of obtaining the eigenvalue c is

P

(2) Without giving the system time to evolve, measure B and then measure C. The probability of obtaining the eigenvalue b and then the eigenvalue c is

P

Assume that in both experiments the system is in state |v

Can the probability P

We may write

P

= ∑_{b}|<v_{c}|w_{b}><|w_{b}u_{a}>|^{2}
+ ∑_{b}∑_{b'≠b}<v_{c}|w_{b}><|w_{b}u_{a}><v_{c}|w_{b'}>*<|w_{b'}u_{a}>*

= P_{a}(b,c) + ∑_{b}∑_{b'≠b}<v_{c}|w_{b}><|w_{b}u_{a}><v_{c}|w_{b'}>*<|w_{b'}u_{a}>*

≠ P_{a}(b,c).

Again we encounter cross terms, leading to interference between different paths.
Just
summing the probabilities for each possible path does not lead to the correct result.
All
the interference effects are then missing. When the intermediate state of the system is
not determined, it is the **probability amplitudes** for the different intermediate
states that must be summed, **not the probabilities.**

**Why?
**The fifth postulate of Quantum Mechanics states that during the measurement of B
in the second experiment, the state of the system abruptly changes from |u

- (i) Always square the
**probability amplitudes**to obtain the probabilistic predictions of Quantum Mechanics. - (ii) When no measurement of an intermediate state is made, always sum the probability amplitudes, not the probabilities.
- (iii) For a system in a linear superposition of states, the probability amplitude is the sum of the partial amplitude. The probability is the square of this sum. The partial amplitudes interfere with each other.