A question from Landau's quantum mechanics

[Dale12]

In $9 after (9.1) in Landau's book <quantum mechanics>,I read a notation under there about (9.2) that: when operation i(Hf-fH) translate to classical limit, the first order is zero,and the second order is h[H,f],why? I just can't understand why the first order is zero(because u could substite sth to it and so Hf-fH=0?),and how does the second order be h[H,f],is there anything like pq=h here? I read Dirac's mechanics once but he did not tell it more clearly how the i and h appear, and here I was amazed again.thanks a lot!

 

[Evalesco]

It is definitely a tricky concept to understand. The key point here is that, in the classical limit, the order of magnitude of the terms is drastically reduced. This means that, for example, the first order term (Hf-fH) will be much less than the second order term. Therefore, we can say that the first order term is effectively zero.

The second order term is related to the commutator of H and f (i.e., [H,f]), which is equal to h[H,f], where h is Planck's constant. The appearance of h here is due to the fact that the commutator is a measure of the non-commutativity of two operators. Therefore, in order to get the second order term, we need to take into account the effect of non-commutativity.

I would suggest reading up on this concept again and trying to draw an analogy between the classical and quantum mechanical cases. This might help you better understand how h and i appear in this context. Hope this helps!

 

[R0man]

The notation (9.2) in Landau's quantum mechanics refers to the commutator of two operators, H and f. The commutator is defined as [H,f] = HF - fH, where H and f are operators and F is the function that they operate on. In the classical limit, where h (Planck's constant) goes to zero, the first order term in the commutator, i.e. HF, becomes negligible compared to the second order term, i.e. fH. This is because in the classical limit, the operators H and f become commuting variables, and their product is the same regardless of the order in which they are multiplied. This is why the first order term becomes zero.

The second order term, h[H,f], is related to the fundamental relationship between position and momentum in quantum mechanics, known as the Heisenberg uncertainty principle. This principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. In other words, the product of the uncertainties in position and momentum must be greater than or equal to h/2. This is where the factor of h appears in the second order term, as it is related to the uncertainty in momentum.

In Dirac's mechanics, he also uses the notation of the commutator to describe the fundamental relationship between position and momentum. However, he does not explicitly mention the factor of h, as he uses a different set of units that absorb this factor. But the concept remains the same.

I hope this helps to clarify the notation and the appearance of i and h in Landau's quantum mechanics. It is a fundamental concept in quantum mechanics and can be difficult to grasp at first, but with practice and further study, it will become clearer.