Confusion about derivative operators

[Loro]

As far as I understand, the momentum operator is:

$\hat{p} = -i \hbar \frac{\partial}{\partial \hat{q}}$

Where I'm not sure at this point if it's mathematically correct to talk about a derivative wrt. the position operator - but the point is, as far as I understand, that this equality is true in general, without taking the Schrödinger representation. It is not a derivative wrt the position eigenvalues.

Since $\hat{p}$ is a Hermitian operator, $\frac{\partial}{\partial \hat{q}}$ must be pure imaginary. So I think it's true that:

$\left( \frac{\partial}{\partial \hat{q}} |ψ> \right)^{\dagger} = <br /> <ψ|\left( \frac{\partial}{\partial \hat{q}} \right)^{\dagger} = <br /> -<ψ| \frac{\partial}{\partial \hat{q}}$

Similarly:

$\hat{H} |ψ> = i \hbar \frac{\partial |ψ>}{\partial t}$

Hence:

$\left( \frac{\partial}{\partial t} |ψ> \right)^{\dagger} = <br /> <ψ|\left( \frac{\partial}{\partial t} \right)^{\dagger} = <br /> - \frac{\partial}{\partial t} <ψ|$

But that is wrong. The minus shouldn't be there. What mistake am I making?
Also, can we say: $\hat{H} = i \hbar \frac{\partial }{\partial t}$ ?

 

[Simon Bridge]

Since $\hat{p}$ is a Hermitian operator, $\frac{\partial}{\partial \hat{q}}$ must be pure imaginary.

... in order to give real eigenvalues you are thinking? But surely you only need any resulting, additional, imaginary components to cancel out?

$\hat{H} = -i\hbar\frac{\partial}{\partial t}$

... kinda, though we'd usually write:
$$\hat{H} = \frac{\hat{p}^2}{2m}$$

So for an energy eigenstate like $\Psi(x,t)=\psi_n(x) e^{iE_nt/\hbar}$ you'll see the two expressions give the same result.

notice that $\frac{\partial}{\partial t}$, in this example, is not purely imaginary yet the Hamiltonian operator is still giving real eigenvalues.

 

[strangerep]

Since $\hat{p}$ is a Hermitian operator, $\frac{\partial}{\partial \hat{q}}$ must be pure imaginary. So I think it's true that:
$$\def\<{\langle}
\def\>{\rangle}
\left( \frac{\partial}{\partial \hat{q}} |ψ\> \right)^{\dagger} =
\<ψ|\left( \frac{\partial}{\partial \hat{q}} \right)^{\dagger} =
-\<ψ| \frac{\partial}{\partial \hat{q}}
$$

Taking adjoints of such operators must be done more carefully. Consider an inner product $C := \<\phi|P \psi\>$, where $P$ is the momentum operator. For $P$ to be selfadjoint we must have $C = \<P \phi|\psi\>$ also.

Writing $C$ in terms of Schrödinger wave functions, etc, (so that $P$ is represented as the derivative you described), this becomes
$$
C := \<\phi|P \psi\> ~\equiv~ -i\hbar \int\! dq \; \phi^*(q) \partial_q \psi(q)
$$
To make the derivative act on $\phi$ instead, we must use integration by parts, together with an assumption that the wave functions vanish sufficiently fast as $q\to\infty$. This gives:
$$
C = + i\hbar \int\! dq \; \Big( \partial_q \phi^*(q) \Big) \psi(q)
~=~ \int\! dq \; \Big(-i \hbar \partial_q \phi(q) \Big)^* \psi(q)
~\equiv~ \<P \phi|\psi\> ~,
$$ as claimed.

HTH.

[Edit: last equation changed after typo pointed out in post#4 below.]

 

[vanhees71]

$$
C = + i\hbar \int\! dq \; \Big( \partial_q \phi^*(q) \Big) \psi(q)
~=~ \int\! dq \; \Big(-i \hbar \partial_q \phi^*(q) \Big) \psi(q)
~\equiv~ \langle P \phi|\psi\rangle ~,
$$ as claimed.

HTH.

This final equation must read
$$C = + i\hbar \int\! dq \; \Big( \partial_q \phi^*(q) \Big) \psi(q)
~=~ \int\! dq \; \Big(-i \hbar \partial_q \phi(q) \Big)^* \psi(q)
~\equiv~ \langle P \phi|\psi\rangle ~,
$$

 

[strangerep]

This final equation must read
$$C = + i\hbar \int\! dq \; \Big( \partial_q \phi^*(q) \Big) \psi(q)
~=~ \int\! dq \; \Big(-i \hbar \partial_q \phi(q) \Big)^* \psi(q)
~\equiv~ \langle P \phi|\psi\rangle ~,
$$

OOPS, yes! I changed my post #3 accordingly. Thanks for the correction.

 

[Loro]

Thanks for the replies.

$+ i\hbar \int\! dq \; \Big( \partial_q \phi^*(q) \Big) \psi(q) <br /> ~=~ \int\! dq \; \Big(-i \hbar \partial_q \phi(q) \Big)^* \psi(q)$

But when you say that, aren't you assuming in this step, that:

$\partial_q ^{ \dagger} = \partial_q$

Which, as I understand, is the result we're trying to justify?

 

[Loro]

Proving this more carefully:

$<\phi | \left( -i\hbar \hat{\partial_q} \right) |\psi > = \int dq' <\phi | q' >< q' | \left( -i\hbar \hat{\partial_q} \right) |\psi > = -i\hbar \int dq' \phi^{*} \partial_{q'} \psi$
$<\phi | \left( -i\hbar \hat{\partial_q} \right)^{\dagger} |\psi > = \int dq' <\phi | q' >< q' | \left( -i\hbar \hat{\partial_q} \right)^{\dagger} |\psi > = +i\hbar \int dq' \phi^{*} \partial_{q'}^{\dagger} \psi$

And demanding that $\hat{p}$ is self-adjoint, both expressions have to be equal.
Hence:

$\int dq' \phi^{*} \partial_{q'}^{\dagger} \psi = -\int dq' \phi^{*} \partial_{q'} \psi$
$\int dq' <\phi |q'><q'| \hat{\partial_{q}}^{\dagger} |\psi> = -\int dq' <\phi | q'><q'| \hat{\partial_{q}} |\psi>$
$<\phi | \hat{\partial_{q}}^{\dagger} |\psi> = - <\phi | \hat{\partial_{q}} |\psi>$

And since this works for any $\phi$ and ψ:

$\hat{\partial_{q}}^{\dagger} = -\hat{\partial_{q}}$

 

[tom.stoer]

Sorry for interrupting but please note that

1) a wave function ψ(q) in q-space or ψ(p) p-space is always a projection of the abstract state |ψ> on the position-eigenstate <q| or the momentum-eigenstate <p|, respectively.
2) the derivative operator ∂_q never acts on bras |ψ> but only on wave functions ψ(q) = <q|ψ>. An expression like ∂_q|ψ> is ill-defined!

 

[dextercioby]

Also, the time derivative 'operator' mentioned in the OP is not an operator in the usual sense, it merely depicts the infinitesimal change suffered by a state of the (rigged) Hilbert space as time passes. It's the "q" or the "p" in the argument of the wavefunction with respect to which we apply the mathematical theory of linear operators in (rigged) Hilbert spaces, not "t". The time is merely a parameter, q and p are the true variables.

Also, in functional analysis there's no derivative of an operator with respect to another operator, it's with respect to a parameter/variable which is a scalar, not a vector of a (rigged) Hilbert space, neither an operator on it.

 

[tom.stoer]

exactly;

and of course

... can we say: $\hat{H} = i \hbar \frac{\partial }{\partial t}$ ?

is wrong; we can't

the time-dep. SE containing the wave functions says that acting with i∂_t on a certain state |ψ> has the same effect as acting with H on that state; but this is not true for arbitrary states, only for solutions of the time-dep. SE.

simple example from algebra: suppose you have a 2-vector v and a 2*2 matrix M; now you can write down an eigenvalue equation (M-λ)v = 0; of course that does not imply that M = λ always holds b/c you can always construct vectors u for which (M-λ)u ≠ 0

 

[Amok]

Also, can we say: $\hat{H} = i \hbar \frac{\partial }{\partial t}$ ?

I really don't think you can. The TDSE states that:

$$H |\psi \rangle = i \hbar \frac{\partial}{\partial t} |\psi \rangle$$

I might be wrong, but I don't think it necessarily implies what you said. In fact, way I see it, what you are suggesting is that the TDSE is:

$$H |\psi \rangle = H |\psi \rangle$$

A completely trivial statement, which carries no interesting physical information. The TDSE is only physically interesting exactly because what you said does not hold. I think you can read the TDSE as "the energy of a state does not change (LHS) when an infinitesimal amount of time goes by" (mayne someone can formulate this better).

 

[Loro]

Thanks.

So one thing I think I've understood:

$<q'| \hat{p} |\psi> = - i \hbar \frac{\partial \psi}{\partial q'}$ holds for any ψ

Whereas:

$<q'| \hat{H} |\psi> = i \hbar \frac{\partial \psi}{\partial t}$ holds only for particular ψ's

And that's why we can't write this expression for the Hamiltonian, but we can for momentum.

2) the derivative operator ∂q never acts on bras |ψ> but only on wave functions ψ(q) = <q|ψ>. An expression like ∂q|ψ> is ill-defined!

But Dirac does it all the time in "Principles of Quantum Mechanics". It would be great if someone more experienced could confirm that, but as far as I understood it - at the beginning of chapter 22 - he defines $\hat{\partial_q}$ such that:

$\hat{\partial_q} |\psi> = | \hat{\partial_q} \psi >$

Where:

$<q' | \hat{\partial_q} \psi > \equiv \partial_q' \psi$

And then he shows, that $\hat{p} = -i \hbar \hat{\partial_q}$

 

[tom.stoer]

let's make a simple example: think about |ψ> as an ordinary 3-vector in R³; think about {<n|} with n=1..3 as a set of orthonormal basis vectors in the dual space which again is R³; then ψ_n = <n|ψ> is the n-th component of |ψ> w.r.t. <n|.

OK, ∂_q has something to do with the q-dependence; in your 3-dim. example we only have a discrete index n instead of a continuous one (q), but that doesn't matter; we are now talking about the n-dependence

the basic fact is that |ψ> does not depend on n, but <n|ψ> does! so changing q slightly and checking how |ψ> changes does not make any sense; let's look at n=1, then at n=2, and let's see how |ψ> changes; of course it doesn't; only ψ_n = <n|ψ> depends on n b/c it is the projection on the n-th direction;

so in QM the vector |ψ> itself does not have an n-dependence

but you are right, |ψ>(t)> carries t-dependence

 

[Amok]

Thanks.

So one thing I think I've understood:

$<q'| \hat{p} |\psi> = - i \hbar \frac{\partial \psi}{\partial q'}$ holds for any ψ

Whereas:

$<q'| \hat{H} |\psi> = i \hbar \frac{\partial \psi}{\partial t}$ holds only for particular ψ's

Yes, the first one is the definition of the momentum operator (in q representation) and the second one is the SE (it is not the definition of the hamiltonian).

 

[Loro]

Ok thanks everyone for explaining to me the bit about the Hamiltonian. I understand the difference now.

Now coming back to the discussion of the $\hat{\partial_q}$

Ok I think we're actually getting closer to my problem.
I understand the argument. Dirac actually says in the book that:

$\hat{\partial_q} |\psi> = 0$

Where $|\psi>$ is the standard ket.
I didn't understand it - but perhaps it is so, for the reasons that you've explained.

But what about this - if in particular: $|\psi> = |q'>$ (a position eigenket)

Then can we still say: $\hat{\partial_q} |q'> = 0$ ?

And I can always expand any $|\psi>$ as a linear combination of $|q'>$'s

 

[tom.stoer]

I know it's bold but I don't agree with Dirac's explanation; it's confusing

 

[Simon Bridge]

I know it's bold but I don't agree with Dirac's explanation; it's confusing

Nothing wrong with that - you can agree or disagree with someone for any reason you like: personal confusion is fine. As long as you don't say that the explanation is wrong on the grounds that you find it too confusing ;)

Nah, that just means it's not the sort of explanation that would be helpful for you. BFD: there are different learning styles. Yours is not a good fit to Dirac's teaching style.

 

[strangerep]

Dirac actually says in the book that:

$\hat{\partial_q} |\psi> = 0$

Where $|\psi>$ is the standard ket.

Where does Dirac say that? If you're looking at his eq(11) on p89, what he
actually writes is
$$
\frac{d}{dq} \psi \rangle ~=~ \frac{d\psi}{dq} \,\rangle
$$
Note that absence of the ``$|$''. Dirac is using his own notation, but the
versions of that notation presented by more modern authors are different.

I checked a few other things in his book, including other parts of his notation.
IMHO, once one understands Dirac's notation properly, nothing is wrong.

@Tom: exactly what part of Dirac's explanation do you think is wrong??

 

[Loro]

Where does Dirac say that?

I'm looking at eqn (12) on page 89, or (23) on page 91. I didn't really get why that is. Maybe it's as tom.stoer described?

And my main problem is actually with (46) on page 94 - that there isn't a minus there, as it would follow from sloppy calculations. This is mind-boggling to me!

 

[Loro]

Coming back to eqn 46 - I've just understood it. I thought there was a contradiction. Thanks everyone for giving me an impulse for thinking about all this more carefully.

I would still love to hear replies about $\hat{\partial_q} |\psi>$ though. It's very interesting, and thought-provoking.

 

[tom.stoer]

@Tom: exactly what part of Dirac's explanation do you think is wrong??

I think acting with derivative operators on kets does not make sense (but I have to check Dirac's book; up to now I only see what is written here in this thread; and this seems to be strange)