I Can Operators Have Multiple Partial Derivatives in Quantum Mechanics?

Happiness
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Suppose Q=2x+t and x=t2, then ∂Q/∂t=1.
But Q can also be written as Q=x+t2+t, then ∂Q/∂t=2t+1.
We now have 2 different answers. But I think there can only be one correct answer.

In reference to the equation in the image, no matter we write Q=2x+t or Q=x+t2+t, <Q> should be the same, so the LHS should be the same. But when we have 2 different answers for ∂Q/∂t, the RHS would not be the same. So we have a contradiction.

Q is an observable of a quantum system. LHS stands for left hand side of the equation.
 

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Happiness said:
then ∂Q/∂t=1

No, because x is also a function of t so it's derivative is not 0.
 
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weirdoguy said:
No, because x is also a function of t so it's derivative is not 0.
∂Q/∂t refers to the partial derivative, not the total derivative. The book goes on to talk about operators that explicitly depend on time (see attached images).

In doing the partial derivative ∂Q/∂t, we do not care about how x depends on t, but only how Q explicitly depends on t.

But no matter how you write Q, <Q> and hence the LHS should be the same, while the RHS are not the same, hence the contradiction.
 

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Happiness said:
In doing the partial derivative ∂Q/∂t, we do not care about how x depends on t, but only how Q explicitly depends on t.
A partial derivative is a derivative with respect to a variable where other variables are held constant. If ##x=t^2##, then ##x## isn't held constant.

Note that this has nothing to do with quantum mechanics. It is calculus. For a given ##f(x,y)##, ##\partial f / \partial y## can have only one solution.
 
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DrClaude said:
A partial derivative is a derivative with respect to a variable where other variables are held constant. If ##x=t^2##, then ##x## isn't held constant.

Note that this has nothing to do with quantum mechanics. It is calculus. For a given ##f(x,y)##, ##\partial f / \partial y## can have only one solution.
Let's consider the Q in footnote 22 (see image).
Q=##\frac{1}{2}##mω2x2, where ω is a function of t.
Then is ∂Q/∂t = mω##\frac{dω}{dt}##x2?

Even though for simple harmonic motion, x is also a function of t (ie. x=x0##\sin##(ωt), where x0 is the amplitude), we do not care how x depends on t when calculating ∂Q/∂t. Is this correct?

I think, in quantum mechanics, it would be more accurate to say <x>=x0##\sin##(ωt). But I don't know whether this point helps in this discussion. After all, it doesn't seem to affect the value of <Q> in my original post, where Q=2x+t, so <Q>=2<x>+t. Is this correct?

(In my original post, instead of x=t2, it would be more accurate to say <x>=t2. But this doesn't seem to change the value of <Q>. So it doesn't seem to resolve the contradiction.)
 

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Happiness said:
Even though for simple harmonic motion, x is also a function of t (ie. x=x0sin(ωt), where x0 is the amplitude), we do not care how x depends on t when calculating ∂Q/∂t. Is this correct?
That's not correct. ##x## is here the coordinate (in position representation; more generically, it would have been better to use here ##\hat{x}##, the position operator). It has no time dependence. For a quantum mechanics oscillator, its position is in its state, that is, in its wave function. What you would have is
$$
\braket{x} = \braket{\psi(t) | \hat{x} | \psi(t)} = x_0 \sin(\omega t).
$$
 
Happiness said:
In doing the partial derivative ∂Q/∂t, we do not care about how x depends on t

Of course we do, otherwise we run into problems, as you can see yourself.
 
First one has to clarify things, because the OP makes no sense to begin with. One should note that in QM ##t## is not an observable and thus not an operator but, of course, the position ##x## is, and it's represented by a self-adjoint operator ##\hat{x}##.

Now in the Schrödinger picture of time evolution (pure) states are represented by time-dependent state vectors ##|\psi(t) \rangle##, which fulfill the Schrödinger equation,
$$\mathrm{i} \hbar \partial_t |\psi(t) \rangle=\hat{H} |\psi(t) \rangle, \qquad(1)$$
where ##\hat{H}## is the Hamilton operator of the system. It's also a self-adjoint operator and represents the total energy of the system. Taking the adjoint of this equation, you get
$$-\mathrm{i} \hbar \partial_t \langle \psi(t) \rangle =\langle \psi(t) |\hat{H}^{\dagger}=\langle \psi(t) | \hat{H}. \qquad (2)$$

Note that in the Schrödinger picture ##\hat{x}## is not dependent on time. So the question is how to define the velocity ##v##, which in classical physics is defined as the time derivative of ##x## along the trajectory of the particle. In QM we like ##\hat{v}## to be a self-adjoint operator for which
$$\langle v \rangle=\frac{\mathrm{d}}{\mathrm{d} t} \langle x \rangle.$$
Now the expectation value for the system in the state represented by ##|\psi(t) \rangle## is given by
$$\langle x \rangle=\langle \psi(t)|\hat{x}|\psi(t) \rangle,$$
and we can calculate the time derivative, using the Eqs. (1) and (2):
$$\frac{\mathrm{d}}{\mathrm{d} t} = (\partial_t \langle \psi(t)|\hat{x}|\psi(t) \rangle + \langle \psi(t) |\hat{x}|(\partial_t \psi(t) \rangle) = \frac{\mathrm{i}}{\hbar} (\langle \psi(t)| \hat{H} \hat{x}|\psi(t) \rangle - \langle \psi(t)|\hat{x} \hat{H}|\psi(t) \rangle=\langle \psi(t)| \mathrm{i}/\hbar [\hat{H},\hat{x}]|\psi(t).$$
Since this should hold true for all ##|\psi(t) \rangle## we conclude that
$$\hat{v}=\frac{\mathrm{i}}{\hbar} [\hat{H},\hat{x}] = \frac{1}{\mathrm{i} \hbar} [\hat{x},\hat{H}],$$
is the operator that represents the velocity. One thus defines a "covariant time dervative" for operators that represent an observable by
$$\mathring{\hat{A}}(\hat{x},\hat{p})=\frac{1}{\mathrm{i} \hbar} [\hat{A}(\hat{x},\hat{p}),\hat{H}].$$
If you have an operator that also explicitly depends on time, then you have of course also the corresponding partial time derivative, i.e.,
$$\mathring{\hat{Q}}(\hat{x},\hat{p},t)
=\frac{1}{\mathrm{i} \hbar} [\hat{Q}(\hat{x},\hat{p}),\hat{H}] + \partial_t \hat{Q}(\hat{x},\hat{p},t).$$
 
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