Schrodinger equation and Heisenberg equation of motion

In summary, the conversation discussed the use of equations in Hatfield's book to reproduce a specific equation (2.36). The difficulty was in plugging equation (2.55) into (2.37) because H is an integral. However, it was pointed out that the operator H is conserved in the Heisenberg picture, so it can be evaluated at any time. The equal-time commutation relations (2.54) were used to evaluate the commutator [H, \phi] in (2.37). The notation and arguments were clarified, and the general formula for commutators of three operators was also mentioned. It was concluded that it is allowed to use t in the integral for H since it doesn
  • #1
Josh1079
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0
My question is that how does the Schrodinger equation arise from the Heisenberg equation of motion in the quantum field formalism.

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These are from Hatfield's book. So I'm having some difficulties to reproduce (2.36) by plugging (2.55) into (2.37) primarily because H is an integral.

Thanks!
 
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  • #2
Just use (2.54). Note that ##\hat{H}## is conserved in the Heisenberg picture, i.e., you can evaluate the integral at time ##t## (it doesn't depend on ##t## after all anyway). That's why you only need the canonical equal-time commutation relations (2.54).
 
  • #3
Hi vanhees, great to see you again! Sorry I don't really follow you this time. I was also trying to use (2.54) and I guess I was stuck because I'm not familiar with an operator that is an integral.

So first of all, for the right side of (2.37) I'm not sure whether [tex] [H, \phi] = (\int dx \phi^* h \phi) \phi - \phi (\int dx \phi^* h \phi) [/tex] or [tex] \int dx \phi^* h \phi \phi - \int \phi dx \phi^* h \phi [/tex], where [tex] h = -\frac{1}{2} \partial^2_x + V(x) [/tex]. Furthermore, when the operators are in the form [tex] h \phi \phi [/tex], is it equal to simply [tex] (h\phi) \phi [/tex] or is it [tex] (h\phi)\phi + \phi (h\phi) [/tex]?
 
  • #4
Wait, I think I get something now.

Is it like this?

[tex] [H, \phi] = H\phi - \phi H = \int dx' \phi^*(x') h \phi(x') \phi(x) - \phi(x) \int dx' \phi^*(x') h \phi(x') [/tex]
[tex] = \int dx' \phi^*(x') h \phi(x') \phi(x) - \int dx' \phi(x) \phi^*(x') h \phi(x') [/tex]
the phi(x) can be taken inside the integral since it's independent of x', then by (2.54),
[tex] = \int dx' \phi^*(x') \phi(x) h \phi(x') - \int dx' \phi(x) \phi^*(x') h \phi(x') = \int \delta (x' - x) h \phi(x') dx' = h \phi(x) [/tex]

Did I make any mistakes in the math?
 
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  • #5
I'm not familiar with your notation, and it's also good to write out the arguments. First of all you have
$$\hat{H}=\int_{\mathbb{R}} \mathrm{d} x \hat{\varphi}^*(t,x) \left (-\frac{1}{2} \Delta +V(x) \right) \hat{\varphi}(t,x).$$
Now ##\hat{H}## is not explicitly time dependent and that implies that it is conserved:
$$\frac{\mathrm{d}}{\mathrm{d} t} \hat{H}=[\hat{H},\hat{H}]+\partial_t \hat{H}=0.$$
That implies that you can use any ##t## in evaluating ##\hat{H}## since ##\hat{H}## doesn't depend on it. Now you have
$$[\hat{H},\hat{\varphi}(t,x)]=\int_{\mathbb{R}} \mathrm{d} x' \left [\hat{\varphi}^*(t,x') \left (-\frac{1}{2} \Delta' +V(x') \right) \hat{\varphi}(t,x'),\hat {\varphi}(t,x) \right ].$$
Now you can just use the equal-time commutator relations given in #1 to show (2.37). You also need the general formula
$$[\hat{A},\hat{B} \hat{C}]=[\hat{A},\hat{C}] \hat{B}+\hat{C} [\hat{A},\hat{B}],$$
valid for any three operators ##\hat{A}##, ##\hat{B}##, and ##\hat{C}##.

It is allowed to use ##t## in the integral for ##\hat{H}## as the time argument of the fields since ##\hat{H}## doesn't depend on time as argued above, and that's why you can use the equal-time commutation relations to evaluate this commutator.
 
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1. What is the Schrodinger equation?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. It is used to calculate the probability of finding a particle in a particular state at a specific time.

2. What is the Heisenberg equation of motion?

The Heisenberg equation of motion is another fundamental equation in quantum mechanics that describes the time evolution of a quantum system. Unlike the Schrodinger equation, it describes the time evolution of the operators that represent physical observables, such as position and momentum.

3. How are the Schrodinger equation and Heisenberg equation of motion related?

The Schrodinger equation and Heisenberg equation of motion are two different mathematical formulations of quantum mechanics. They are equivalent and can be used interchangeably to describe the time evolution of a quantum system. However, the Schrodinger equation is more commonly used for non-relativistic systems, while the Heisenberg equation is useful for describing relativistic systems.

4. What is the difference between the Schrodinger equation and the Heisenberg equation of motion?

The main difference between the Schrodinger equation and the Heisenberg equation of motion is that the Schrodinger equation describes the time evolution of the wave function of a quantum system, while the Heisenberg equation describes the time evolution of the operators that represent physical observables. Another difference is that the Schrodinger equation is a partial differential equation, while the Heisenberg equation is a set of linear equations.

5. Why are the Schrodinger equation and Heisenberg equation of motion important in quantum mechanics?

The Schrodinger equation and Heisenberg equation of motion are important because they are the fundamental equations used to describe the behavior of quantum systems. They allow us to make predictions about the behavior of particles at the quantum level, and have led to many groundbreaking discoveries in physics, including the development of modern technologies such as transistors and lasers.

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