Displacement Operator: Explaining Dirac's Equality

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In summary: In summary Dirac's text discusses how the displacement operator is related to the exponential and how the two operators, D and I, are related. If y tends to zero then the taylor expansion of the exponential becomes exact and the equality ⁡[D*exp⁡(iy)-1] = [D-1+iy] holds good.
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bikashkanungo
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In Dirac's text regarding displacement operator it is given that :-
lim(δx→0)⁡[D*exp⁡(iy)-1]/δx =lim(δx→0) [D-1+iy]/δx = dx + iax
Where dx = displacement operator =lim(δx→0) [D-1]/δx
ax = lim(δx→0) y/ δx
and it is assumed that y tends to zero as δx tends to zero
can anyone explain how the equality of ⁡[D*exp⁡(iy)-1] = [D-1+iy] holds good ??
 
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  • #2
bikashkanungo said:
In Dirac's text regarding displacement operator it is given that :-
lim(δx→0)⁡[D*exp⁡(iy)-1]/δx =lim(δx→0) [D-1+iy]/δx = dx + iax
Where dx = displacement operator =lim(δx→0) [D-1]/δx
ax = lim(δx→0) y/ δx
and it is assumed that y tends to zero as δx tends to zero
can anyone explain how the equality of ⁡[D*exp⁡(iy)-1] = [D-1+iy] holds good ??

If y tends to zero then the taylor expansion of the exponential becomes exact. I think that's what he's getting at. Although if this were the case the limit would be D iy +D-1. Did you leave out parentheses?
 
  • #3
@nileb : No its exactly as given in Dirac's book , I did not leave out any paranthesis
 
  • #4
I'm fairly sure there should be a D in front of the iy and iax. Perhaps a misprint? Does he use the formula ever again?
 
  • #6
I think Fitzpatrick copy-pasted from Dirac...Anyway, Dirac's unrigorous treatment looks quite dubious.
 
  • #7
Hi! I was reading the Dirac's text and I was very disappointed for his treatments of the infinitesimal displacement operator, so I found this post and I want to add a reply. I don't know if what I want to say is right but is my interpretation of the equality that appears in dirac's book:

lim(δx→0)⁡[D*exp⁡(iy)-I]/δx =

Obs. I is the Identity operator

lim(δx→0) exp(iy)*[D-exp(-iy)*I]/δx =

Obs. now, in such expression I can say that the exp(iy) in front of all is an arbitrary phase factor and so is completely irrelevant (the important thing is the relative phase factor between the two operators D and I that appear in the equation)

lim(δx→0) [D-exp(-iy)*I]/δx =

lim(δx→0) [D-I+iy*I]/δx =

dx + ax

The thing that I don't understand yet is the taylor expansion for the exponential, I don't understand why he assume y → 0 as δx→0

Please anyone can write me back to tell me if I'm completely wrong?? Thanks;)
 

FAQ: Displacement Operator: Explaining Dirac's Equality

What is a displacement operator?

A displacement operator, also known as a translation operator, is a mathematical operator used in quantum mechanics to describe the motion of particles in space. It represents the change in position of a particle from one point to another.

How does a displacement operator relate to Dirac's equality?

Dirac's equality, also known as the fundamental commutation relation, states that the position and momentum operators do not commute. This means that the order in which they are applied matters. The displacement operator is used to describe the change in position of a particle, which is directly related to its momentum, thus making it a crucial component in understanding Dirac's equality.

What is the mathematical formula for a displacement operator?

The mathematical formula for a displacement operator is D(x) = e^(-ixp/hbar), where x represents the displacement in position, p represents the momentum, and hbar is the reduced Planck's constant.

How is a displacement operator used in quantum mechanics?

In quantum mechanics, a displacement operator is used to calculate the probability amplitude of a particle at a given position, taking into account its initial position and momentum. It is also used in the quantum harmonic oscillator to describe the motion of a particle in a potential well.

What are some real-world applications of displacement operators?

Displacement operators have many real-world applications, such as in quantum computing, where they are used to manipulate qubits and perform operations on quantum states. They are also used in quantum cryptography and quantum teleportation, as well as in quantum simulations of chemical reactions and materials. In addition, displacement operators have applications in signal processing and image processing.

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