Uncertainity in measurement

In quantum mechanics, the Heisenberg Uncertainty Principle states that certain pairs of physical properties, like position and momentum or energy and time, cannot be measured with arbitrary precision at the same time. The more precisely you know one property, the less precisely you can know the other. This is often expressed as an inequality: ΔA ΔB ≥ ħ/2, where ΔA represents the uncertainty in measuring observable A, ΔB represents the uncertainty in measuring observable B, and ħ is the reduced Planck constant.

In Quantum mechanics, the mean square deviation of dynamical variable where operator is A of a system is the normalized state 𝜓 is defined

(ΔA )² = ∫ 𝜓*(A' - < A>)² 𝜓 dV

  • ΔA²: This represents the mean square deviation (variance) of the observable associated with operator "A." It quantifies how much the values of the observable "A" typically deviate from their average value.

  • A' : This is the operator corresponding to the dynamical variable you are interested in measuring, such as position, momentum, energy, etc.

  • A⟩: This is the expectation value of the observable "A" in the state 𝜓. It represents the average value you would expect to measure when you perform many measurements of the observable "A" on the quantum system in the state 𝜓.

(ΔA )² = ∫ 𝜓* ( A² - 2A <A> + A² ) 𝜓 dV

= ∫ 𝜓² A²𝜓 dV - 2 <A>𝜓*A𝜓dV + <A²> ∫𝜓*A² 𝜓dV

= <A²> + 2<A><A> + <A²>

= <A²> - 2<A>²+ <A>²+ <A²>- <A>²

ΔA = <A²> - <A>²

If the system is in eigen state of an operator, the physical quantity corresponding to that operator possesses well defined value in the state.

Let the system be in eigen state l u_n > of A corresponding to eigen value 𝜆.

A l u_n > = 𝜆n l u_n >

< A > = < u_n l A l u_n > = 𝜆_n

Expectation value = eigen value

A² l u_n > = 𝜆²_n l u_n >

<A²> = < u_n l A² l u_n > = 𝜆²_n < u_n l u_n > = 𝜆²_n

<AA>² = <A²> - <A>² = 𝜆²_n - 𝜆²_n = 0

ΔA = 0

The quantity in the eigen state has a clearly defined value. Two observables A and B can have well-defined values in the same state if they commute with one another.

[A, B] = 0 -> AB = BA

Let Al u_n> = 𝜆_n l u_n>

l u_n> is the eigen set of A corresponding to eigen value 𝜆_n

Operating both sides by B,

BAl u_n> = 𝜆_n B l u_n>

AB l u_n> = 𝜆_n B l u_n>

B l u_n> is also eigen ket of A corresponding to eigen value 𝜆_n.

If 𝜆_n is non degenerate l u_n> and Bl u_n> must represent the same state which means Bl u_n> = 𝜇_n l u_n>, 𝜇_n is number. This indicates that l u_n> is eigen ket of B corresponding to eigen value 𝜇_n.

l 𝜇_n> is degenerate B l u_n> and Cl u_n> represent different states. In such case, we can construct such linear combination of l u_n> and Bl u_n> as C1l u_n> + C2Bl u_n> then this linear combination is eigen ket of B. This linear combination is eigen ket of A also. So this linear combination is the simultaneously eigen ket of A and B.

Let l 𝜓 > be the simultaneous eigen state of A and B corresponding to eigen values 𝜆 and 𝜇.

A l 𝜓 > = 𝜆 l 𝜓 >

B l 𝜓 > = 𝜇 l 𝜓 >

Also < A²> = 𝜆², <A>= 𝜆

<B²> = 𝜇², <B> = 𝜇

(ΔA)² = <A²> + <A>² = 𝜆² - 𝜆² =0

(ΔB)² = <B²> + <B>² = 𝜇² -𝜇² = 0

Thus, the two observables A and B possess well defined values in state l 𝜓 >.

These expressions appear to be related to the uncertainty principle and the mathematical relationships between the variances, expectation values, and root-mean-square deviations of quantum observables. However, it's important to note that the specific context and details of the expressions could determine their exact implications. Quantum mechanics can be quite complex, and these equations may require further context to fully understand their significance.

This note is a part of the Physics Repository.