Given creation and annihilation operators, ${a^{\dagger}(x,t)}$ and $a(x,t)$ in non-relativistic quantum field theory, respectively, which satisfy the following properties:

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Now, I want to prove $$[H,N]=0.\tag{1}$$

I've used 2 ways to prove it. One is consider this equation using Heisenberg equation of motion, which is quite straightforward. However, when i try to prove it using commutation relations above, some problems show up. Since $$H=\int{d^3x \space a^{\dagger}(x,t)\nabla^2_xa(x,t)},\tag{2}$$ where $\nabla^2_xa$ indicate that the $\nabla^2$ operator acts on variable $x$.

And the number operator is defined as $$N=\int{d^3x \space a^{\dagger}(x,t)a(x,t)}.\tag{3}$$ My proof goes follows:

$$HN=\int{d^3x \space a^{\dagger}(x,t)\nabla^2_xa(x,t)} \int{d^3x' \space a^{\dagger}(x',t)(x',t)} =\int{d^3xd^3x' \space a^{\dagger}(x,t)\nabla^2_xa(x,t) a^{\dagger}(x',t) a(x',t)} \tag{4}$$ Next, i am using the commutation relation: $$[a(x,t),a^{\dagger}(x',t)]=\delta^{(3)}(x-x').\tag{5}$$ This leads to: $$HN=\int{d^3xd^3x' \space a^{\dagger}(x,t)\nabla^2_x (a^{\dagger}(x',t) a(x,t)+\delta^{(3)}(x-x')) a(x',t)}\tag{6}$$

If I regard $\nabla_x$ and $a^{\dagger}(x',t)$ commute, and perform the delta function integral, this ends up with 2 parts. So the equation equals to: $$NH+H.\tag{7}$$ How am i going to proceed properly? Where did i make mistake?

I deal with the delta function part like this:

Part of the $HN$ is: $$\int{d^3xd^3x' \space a^{\dagger}(x,t)\nabla^2_x \delta^{(3)}(x-x') a(x',t)}\tag{8}$$ Integating over $dx'$ gives:

$$\int{d^3x \space a^{\dagger}(x,t)\nabla^2_xa(x,t)}\tag{9}$$ So this is obviously the Hamiltonian. What's wrong with this?


Hint: This simplest way to prove eq. (1) is to use the formulas $$[a(x),N]~=~a(x)\qquad\text{and}\qquad [N, a^{\dagger}(x)]~=~a^{\dagger}(x)$$ directly in eq. (2), and that the number operator $N$ commutes with the derivative $\nabla_x^2$.

(In this way we avoid dealing with derivatives of the Dirac delta distribution. We should stress that the latter is also a feasible route with appropriate care.)

  • $\begingroup$ Thanks for pointing that out! Following your hints I get it done. However, I am still confused why my way doesn't work. $\endgroup$ – Universe Maintainer Feb 3 at 14:13
  • $\begingroup$ FWIW, eq. (6) is fine but eq. (7) is not. $\endgroup$ – Qmechanic Feb 3 at 14:47
  • $\begingroup$ Thank you! but i still haven't figure out how to deal with the delta function part. $\endgroup$ – Universe Maintainer Feb 4 at 10:39
  • $\begingroup$ I updated my derivation of the delta function part. Could you please tell me what's wrong with it? $\endgroup$ – Universe Maintainer Feb 4 at 14:56
  • $\begingroup$ FWIW, eq. (8) & (9) are fine. $\endgroup$ – Qmechanic Feb 4 at 15:28

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