Solving Commutator [E,x]: Stuck on One Tiny Portion

[Geezer]

Okay, I *know* that E and x are supposed to commute, but I'm stuck on one tiny portion when I work through this commutator...

So, here's my work. Feel free to point out my error(s):

$$[E,x]\Psi=(i\hbar\frac{\partial}{\partial t}x-xi\hbar\frac{\partial}{\partial t})\Psi$$

...which becomes...


$$[E,x]\Psi=(i\hbar\frac{\partial x}{\partial t}\Psi+i\hbar x\frac{\partial\Psi}{\partial t}-i\hbar x\frac{\partial\Psi}{\partial t})$$

So, here's my question: must $$\frac{\partial x}{\partial t}\Psi$$

necessarily equal zero? Clearly it should if the commutator is going to be equal to zero...

Also, I recognize that something like dx/dt in quantum is fairly meaningless, but dx/dt is a lot lot <p>/m, which isn't meaningless in quantum...

Feel free to set me straight...

 

[Pengwuino]

First of all, they do not commute (think simple harmonic oscillator). Second, X is not a function. Psi is the function you're acting upon. Third, as one can find in a previous thread on this topic, https://www.physicsforums.com/showthread.php?t=326154 , the time derivative is not a valid operator so commutator in this form makes no sense.

 

[Hurkyl]

So, here's my question: must $$\frac{\partial x}{\partial t}\Psi$$

necessarily equal zero?

Yes. Because $\partial x / \partial t = 0$.

 

[Hurkyl]

the time derivative is not a valid operator so commutator in this form makes no sense.

However, the time derivative is a valid partial operator on the set of Hilbert-space-valued functions of t.

 

[Geezer]

I posted this last night before going to bed. While brushing my teeth, it occurred to me that $$i\hbar\frac{\partial}{\partial t}$$ is the energy operator associated with the time independent Schrödinger equation; if $$\Psi$$ is "time independent," then its derivative WRT time must be zero.

 

[Geezer]

First of all, they do not commute (think simple harmonic oscillator). Second, X is not a function. Psi is the function you're acting upon. Third, as one can find in a previous thread on this topic, https://www.physicsforums.com/showthread.php?t=326154 , the time derivative is not a valid operator so commutator in this form makes no sense.

What's your take on this (link below)? They seem to say that they do commute...

http://quantummechanics.ucsd.edu/ph130a/130_notes/node116.html#example:comEx"

 

[kuruman]

Please set me straight. I always thought that E is a constant not an operator. Starting from the Schrödinger equation

$$i \hbar\frac{\partial \Psi }{\partial t} = H \Psi$$

one observes that the partial diff. eq. separates by assuming a solution

$$\Psi(x,t) = \psi (x)e^{- i E t/\hbar}$$

where E is the separation constant. In that case, we get

$$H\psi = E \psi$$

but this does not mean that H = E. Looking at the Schrödinger Equation, it seems to me that the correct way to write the operator is

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

So [E, x] = 0, but [H, x] may or may not be zero.

 

[Geezer]

First of all, they do not commute (think simple harmonic oscillator).

But $$i\hbar\frac{\partial}{\partial t}$$ is not the full energy for the harmonic oscillator, because the harmonic oscillator is subject to a potential (i.e. V=1/2 kx^2), so this commutator, applied to the harmonic oscillator, doesn't make sense...

(Folks, feel free to let me know if I'm on the right track or not...)

 

[Geezer]

but this does not mean that H = E. Looking at the Schrödinger Equation, it seems to me that the correct way to write the operator is

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

So [E, x] = 0, but [H, x] may or may not be zero.

That's what I'm thinking. The Hamiltonian incorporates a potential energy, too...

 

[Hurkyl]

I do not believe defining

$$\hat{E} = i \hbar \frac{\partial}{\partial t}$$​

to be that uncommon. If nothing else, you would need that definition for relativistic QM, because energy is the time-component of the momentum 4-vector.

The equation

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

is definitely wrong. The equation

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

is an equation we are seeking to solve -- it is not an identity of operators and vectors.

 

[Geezer]

I do not believe defining

$$\hat{E} = i \hbar \frac{\partial}{\partial t}$$​

to be that uncommon. If nothing else, you would need that definition for relativistic QM, because energy is the time-component of the momentum 4-vector.

The equation

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

is definitely wrong. The equation

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

is an equation we are seeking to solve -- it is not an identity of operators and vectors.

Yes, I'll agree that H is not an operator, but we often do commutation relations with the Hamiltonian (or, at least with the kinetic energy operator)...

 

[Hurkyl]

Yes, I'll agree that $$\hat{H}$$ is not an operator

Was that a typo?

 

[Geezer]

Was that a typo?

Whoops...yes, that was a typo. Sorry. I'll correct it.

 

[kuruman]

I do not believe defining

$$\hat{E} = i \hbar \frac{\partial}{\partial t}$$​

to be that uncommon. If nothing else, you would need that definition for relativistic QM, because energy is the time-component of the momentum 4-vector.

The equation

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

is definitely wrong. The equation

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

is an equation we are seeking to solve -- it is not an identity of operators and vectors.

You have set me straight. Thanks.

 

[Geezer]

Please set me straight. I always thought that E is a constant not an operator. Starting from the Schrödinger equation

But I've seen E used as an operator, like here: http://quantummechanics.ucsd.edu/ph130a/130_notes/node116.html#example:comEx
and I remember using it as an operator (or "operator equivalent") in class, too.

 

[turin]

Suppose that |x> and |y> are generic position vectors. I assume that the position vectors span the extended Hilbert space. So, what is <x|E|y>? Do all of these matrix elements vanish? If so, then I don't think that E is a valid operator (in ordinary nonrelativistic QM). What is the context of the problem? Is it relativistic QM? If so, then I don't understand the meaning of x as an operator.

 

[Hurkyl]

Suppose that |x> and |y> are generic position vectors. I assume that the position vectors span the extended Hilbert space. So, what is <x|E|y>? Do all of these matrix elements vanish? If so, then I don't think that E is a valid operator (in ordinary nonrelativistic QM). What is the context of the problem? Is it relativistic QM? If so, then I don't understand the meaning of x as an operator.

E is not an operator on kets. E is a (partial) operator on ket-valued functions on R.

If we use |y> to denote the constant (generalized-)ket-valued function, then <x|E|y> is, indeed, zero-distribution-valued constant function. But the |y> do not span the space of all (generalized-)ket-valued functions.

 

[turin]

I don't think that I understood what you wrote. Some ideas just popped into my head, though. If you don't mind, I would appreciate your comments.

I'm guessing that R is the space of the time parameter. Then, it seems like there exists a different Hilbert space for each value of time. When a Hilbert space operator, such as position or momentum, acts on a Hilbert space, the matrix elements are taken between two elements of that same space. However, E characterizes the difference between two Hilbert spaces, and the matrix elements of E are taken between elements of two different Hilbert spaces. With that said, I am even more confused about the meaning of [E,X], because it can only act in one direction consistently. For instance, E and X can both act to the right on the Hilbert space at t. But then E must act to the left on the Hilbert space at t+dt, while X acts to the left on the Hilbert space at t. So, it seems that the matrix element of this commutator, <ψ_f|[E,X]|ψ_i>, does not make sense, because one of |ψ_i> or |ψ_f> must be a member of two different Hilbert spaces at the same time. I guess I'm wrong.

Another thought that I had was that I can make an arbitrary change of basis: |x>→e^if(x,t)|x>, where I will choose f(x,t) to be real-valued for simplicity of html tags. Then, <y|E|x>→-(∂f/∂t)e^{i(f(x,t)-f(y,t))}<y|x>→-(∂f/∂t)δ(x-y). This can be nonzero, as far as I can tell. However, E is probably represented differently in this new basis, in just such a way that <y|E|x> still vanishes, and what I just wrote is probably incorrect.

 

[Hurkyl]

I think you might be overcomplicating things.

What you're describing is a sort of differential-geometric viewpoint on time-varying kets. While I'm sure that is a reasonable thing to do, it's going to be a lot more complicated, and I'm not sure what benefit you'd get.


If $\psi$ denotes a time-varying ket... that is a function $\mathbb{R} \to \mathcal{H}$, then:

For each t, $\psi(t)$ is a ket to which we can¹ apply X. So $X \psi$ makes perfect sense as a time-varying ket: specifically

$(X \psi)(t) = X (\psi(t))$​

and so we can use X as an operator not only on kets, but also on time-varying kets.

It makes no sense to try and apply E to a ket, because it is an operator on time-varying kets:

(E \psi)(t) = i \hbar \psi'(t)​

X and E both make sense as operators acting on time-varying kets. Since $X \psi$ is a time-varying ket, $E X \psi$ makes sense. Similarly, $X E \psi$ makes sense, as does $[E,X]\psi$.

By considering conjugation, one can make similar statements for them as acting on time-varying bras.

In short, applying X makes sense, because they are ket-valued, and applying E makes sense because they are functons of time.



If you're going to stick with the differential-geometric picture, you're going to have to put in place some extra scaffolding.

You've replaced time-varying kets with "ket fields" over R.
$\partial / \partial t$ is going to be an ordinary tangent vector field on R.
X is an operator field -- essentially the same thing as a rank (1,1) tensor. (But as applied to kets, not as to tangent vectors)

But you need a new thing -- there is some sort of connection on your vector bundle. This is an object that takes a ket field and a tangent vector field, and returns something that captures the idea of a "directional derivative" of the ket field along the tangent vector field. E would be the differential operator formed by feeding $\partial / \partial t$ into that connection (multiplied by the appropriate constant).

1: I'm neglecting issues about the domain of X here.

 

[turin]

OK, I understand now to my level of satisfaction. Thank you very much, Hurkyl! However, now the commutator seems trivial; basically like a commutator of a c-number (i.e. that it is simply defined to vanish, and the only reason I was confused was that I was trying to prove a definition).

I think you might be overcomplicating things.

Most likely.

... $X E \psi$ makes sense, ...

That one takes a little bit of convincing on my part, but I think I can accept that. I am thinking of Eψ(t) as proportional to the sum of two Hilbert space vectors: the vector ψ(t+δt) and the vector inverse of ψ(t), with part of the proportionality factor being 1/δt. I argue to myself that this sum must also be a Hilbert space element (because the Hilbert space is a vector space), so X is allowed to act on Eψ(t). The only thing that I am a little bit uneasy about in your whole argument, then, is convincing myself that the limiting procedure δt→0 does not disturb the proportionality. I guess I could argue physically that this limit must be well-defined (but then I worry about weird things like wave-function collapse). I believe that's just something that I need to think about (although your insight is thouroughly appreciated)!.

But you need a new thing -- there is some sort of connection on your vector bundle.

I suspected as much.

 

[Hurkyl]

I am thinking of Eψ(t) as proportional to the sum of two Hilbert space vectors: the vector ψ(t+δt) and the vector inverse of ψ(t), with part of the proportionality factor being 1/δt.

Right(ish) -- the ordinary definition of the derivative of a vector-valued function on R works no matter what the target vector space is. E is given by
$$(E \psi)(t) = i \hbar \lim_{h \to 0} \frac{\psi(t+\epsilon) - \psi(t)}{\epsilon}$$

I argue to myself that this sum must also be a Hilbert space element (because the Hilbert space is a vector space), so X is allowed to act on Eψ(t).

And like ordinary derivatives, some functions won't be differentiable. And some of those times, the derivative will exist in a larger space -- e.g. a time-varying wavefunction might be time differentiable, but the derivative might not be square-integrable.

The operators P and X have the same issues, even without involving time-varying stuff. $X\psi$ and $P\psi$ are undefined for many kets. And sometimes, they will be defined for wavefunctions, but the result won't be square-integrable.


Anyways, it is true that since this derivative appears in the Schrödinger equation, any non-differentiable time-varying ket cannot represent the time evolution of a quantum system.




By the way, I would like to retract an earlier statement -- it turns out that for a time-varying bra $\langle \psi |$, $\langle \psi | E$ is almost never a time-varying bra. Fortunately, we don't actually have to worry about what kind of object it is, as long as we focus on how E applies to time-varying kets.