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I have just started reading through quantum mechanincs (Griffiths), where he says that the below boundary term is zero, as $\Psi$ must go to zero at infinity (when calculating $d \langle x \rangle / dt$) to simplify and do the integration by parts:

$$x\Psi^* \frac{\partial \Psi}{\partial x} - x\frac{\partial \Psi^*}{\partial x}\Psi \ \Bigg|_{-\infty}^{+\infty} = 0$$

At infinity, well, $x$ also goes to infinity making the limit indeterminate. I don't really understand this claim. Can anyone tell me what I'm overlooking here?


Edit1:

So, to be more precise on where my confusion is, we start with

$$\frac{d}{dt} \int_{-\infty}^{+\infty} x(\Psi^*\Psi) dx = \frac{i \hbar}{2m} \int_{-\infty}^{+\infty} x \frac{\partial}{\partial x}\big( \Psi^* \frac{\partial \Psi}{\partial x} - \frac{\partial \Psi^*}{\partial x} \Psi \big)dx$$ which is the result of subbing $\partial\Psi$ for what it is in Schrödinger's equation. From this, we expand it (using integration by parts), resulting in

$$\frac{d \langle x \rangle}{dt} = -\frac{i \hbar}{2m} \int_{-\infty}^{+\infty} \frac{\partial}{\partial x}\big( \Psi^* \frac{\partial \Psi}{\partial x} - \frac{\partial \Psi^*}{\partial x} \Psi \big)dx + \Big(x\Psi^*\frac{\partial \Psi}{\partial x} - x \frac{\partial \Psi^*}{\partial x}\Psi\Big)_{-\infty}^{+\infty}$$

and this is where they argue that the boundary term is zero. I can't see why because that term also has $x$, making the limit indeterminate $0 \times \infty$.

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  • $\begingroup$ Although you started with $\frac{d\langle x\rangle}{dt}$, there is no $dx/dt$ in the integral $\langle p\rangle=\int^{\infty}_{- \infty} \Big(x\Psi^* \frac{\partial \Psi}{\partial x} - x\frac{\partial \Psi^*}{\partial x}\Psi\big)dx$ - assuming this is the integral. You would need to rework it so it looks like $\langle p\rangle=\int^{\infty}_{- \infty} \Psi^* \frac{\hbar}{i} \frac{\partial }{\partial x}\Psi dx$ using the Schrodinger equation - the gymnastics are roughly the same. $\endgroup$ – Cinaed Simson Dec 13 '19 at 8:38
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    $\begingroup$ I understand how this happens, my question is the expansion that follows which I wrote in detail in "edit 1". $\endgroup$ – cutus_low Dec 13 '19 at 19:28
  • $\begingroup$ The $x$'s should be sandwiched between the $\Psi$'s - and or their derivative - which are square integrable functions, and by definition, vanish at infinity. $\endgroup$ – Cinaed Simson Dec 13 '19 at 21:59
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    $\begingroup$ @CinaedSimson There's no distinction between the two integrals you've written, when we consider the position basis representation of operator $\hat{x}$: $x \Psi^*(x,t) \Psi(x,t) = \Psi^*(x,t) x \Psi(x,t)$. It is just the multiplication of three functions $x$, $\Psi^*(x,t)$, and $\Psi(x,t)$. $\endgroup$ – Ajay Mohan Dec 17 '19 at 7:40
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    $\begingroup$ @CinaedSimson To clarify the matter further: $|\Psi \rangle = \int \Psi(x,t) | x \rangle dx \Rightarrow \hat{x} | \Psi \rangle = \int \Psi'(x,t) | x \rangle dx$, where $\Psi'(x,t)=x\Psi(x,t)=\Psi(x,t)x$. If instead the momentum operator had acted on it, we would have had: $\hat{p}| \Psi \rangle = \int \Psi''(x,t) | x \rangle dx$, where $\Psi''(x,t)= -i\hbar \frac{\partial}{\partial x} \Psi(x,t)$. $\endgroup$ – Ajay Mohan Dec 17 '19 at 7:47
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In QM while doing these manipulations you always assume that the wave function is square integrable i.e. in $L^2(\mathbb R)$ space. Now mathematically speaking functions in the $L^2(\mathbb R)$ space can be wild. So rather than working with $L^2(\mathbb R)$ functions, we usually restrict ourselves to Schwartz functions $\mathcal S(\mathbb R)$, which are functions that satisfy

$$|f^{(n)} (x) \cdot x^m| < \infty \quad \text{for } x \to \infty \qquad \forall m,n $$

in other words Schwartz functions are functions, which fall faster than any power, regardless of how many derivatives you take. For example, $1/(1+x^2)$ on $\mathbb R$ is not a Schwartz function because

$$\frac{x^m}{1+x^2} \to \infty \quad \forall m>2$$

Note that $1/(1+x^2)$ is square integrable but not Schwartz. Another example would be $e^{-x^2}$ which you can show with a little bit of calculus is indeed a Schwartz function. Even though Schwarz functions look very restrictive, they are dense in $L^2$ norm, that is you can approximate the integral of any $L^2$ function with a sequence of Schwartz functions.

In physics we usually make the assumption that the wave function falls off "fast enough" and usually "fast enough" means being a Schwartz function. There is also an expectation that solutions to Schrödinger equation with nice enough potentials will be Schwartz functions. This is for example true for the harmonic oscillator or the hydrogen atom.

If we assume that the solutions are Schwartz functions then, you can easily see why you can drop terms like $x |\Psi|^2$ as $x \to \infty$.

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  • $\begingroup$ This is interesting. Usually it's said that any normalizable (i.e. square-integrable) wavefunction is a completely "physically reasonable" state. Does this mean that there are normalizable wavefunctions that can never represent valid physical states? $\endgroup$ – Sahand Tabatabaei Dec 17 '19 at 2:33
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    $\begingroup$ @SahandTabatabaei I mean these are words that people use while not being careful about what they could potentially mean. You can have a no-where differentiable function, which is square integrable. Does that mean this is a possible wave function? In my opinion not really. $\endgroup$ – Gonenc Dec 17 '19 at 2:40
  • $\begingroup$ I see. I assume there is a way to show that such pathological examples don't exist in Schwartz space? In other words, how do we know that we don't need to restrict our functions even more than Schwartz functions? I don't even know what I mean by "pathological" here. What would the necessary and sufficient conditions be for a wavefunction to be "valid"? Is it just the Schwartz space by definition? $\endgroup$ – Sahand Tabatabaei Dec 17 '19 at 2:46
  • $\begingroup$ @SahandTabatabaei I think your question is either a philosophical one or a PDE question (or a combination thereof). First of all, Schwartz functions are by definition infinitely differentiable, die off really quickly and are square-integrable so a "physicists dream" if you want. The question is, whether the PDE $H \psi = i \partial_t \psi$ has a solution in the schwartz space, which is a question of PDE. If it doesn't, and whether or not the (semi-pathological) solution is "physical" depends on your philosophy position and/or the physical application. $\endgroup$ – Gonenc Dec 17 '19 at 2:58
  • $\begingroup$ "Usually it's said that any normalizable (i.e. square-integrable) wavefunction is a completely "physically reasonable" state." Nonsense! A wavefunction extending over a trillion light years would be completely unreasonable. Yet even that pales in comparison to the length of $\mathbb{R}$. $\endgroup$ – knzhou Dec 17 '19 at 4:24
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A linear operator $A$ in a Hilbert space $\cal H$, is defined as a linear map associating elements of $\cal H$ to elements of $\cal H$.

The key information you need is that not every possible linear map having the whole $\cal H$ as domain is an operator if the Hilbert space is infinite dimensional (like $L^2(\mathbb R)$). There is a special set of linear operators (the bounded or continuous operators) mapping bounded sets into bounded sets, which can be defined over the whole Hilbert space, but other operators exist (the unbounded ones) whose domain is a proper subset of $\cal H$ and cannot be extended to the whole Hilbert space.

Position and momentum operators are important examples in QM of unbounded operators. In the case of the usual representation: $$ \begin{eqnarray} \hat x &:& \hat x \psi(x,t) \rightarrow x \psi(x,t) \\ \hat p &:& \hat p \psi(x,t) \rightarrow -i \hbar \frac{d}{dx}\psi(x,t). \end{eqnarray} $$ In the case of position, its domain $D_{\hat x}$ has to be the proper subset of the $L^2(\mathbb R)$ space whose elements are such that $x\psi(x,t) \in L^2(\mathbb R)$. In the case of momentum, $D_{\hat p}$ coincides with the subset of square integrable and differentiable functions.

Every time one is manipulating expressions containing unbounded operators, it is important to identify the underlying domain, if a precise mathematical meaning has to be given to the expressiones.

Coming to your expression, the starting point (expectation value of $\hat x$) requires that the wavefunction $\Psi(x,t)$ would be in $D_{\hat x}$. In order to exploit the fact that $\Psi(x,t)$ is solution of the time dependent Schrödinger equation, $\Psi(x,t)$ has also be in the domain $D_{\hat p^2}$, which is the subset of twice differentiable elements of $L^2(\mathbb R)$). Therefore, the whole formal manipulation implicitly assumes $\Psi(x,t) \in D_{\hat p^2} \cap D_{\hat x}$.

Vanishing of your expression is ensured in such a domain by Schwarz's inequality. Indeed, $$ \left| \int_{-\infty}^{+\infty} x\Psi^* \frac{d \Psi}{dx} dx \right| \le \sqrt{\int_{-\infty}^{+\infty} x^2\left|\Psi\right|^2 dx} \sqrt{\int_{-\infty}^{+\infty} \left|\frac{d \Psi}{dx} \right|^2 dx}. $$ Both the integrals on the right hand side of the previous inequality exist and are finite for any $\Psi \in D_{\hat p^2} \cap D_{\hat x}$. Therefore the integral on the left hand side is finite and consequently the integrand function must go to zero when approaching $-\infty$ and $+\infty$

Using dense sets of nice-behaving functions, like in Gonenc' answer is a different way to arrive to the same conclusion, although, in my opinion, it obfuscates the key role played by the unbounded operators which are at the starting point of your expression.

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  • $\begingroup$ I think yours is a nice answer, which any physicist would agree with. However I think you can easily construct functions in $D_{\hat p^2} \cap D_{\hat x}$, which don't even have a limit as $x \to \pm \infty$. Hence, the reason for passing to Schwartz functions. $\endgroup$ – Gonenc Dec 17 '19 at 20:26
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    $\begingroup$ @Gonenc I agree that Schwartz functions may be a convenient domain to simplify the proof. However, on the one hand, it is not obvious that it is the largest domain allowing to proof the result. On the other hand, it remains that the only justification for restricting the set of wavefunctions is to ensure that they are elements of a subset of the intersection of the domains of the relevant operators. $\endgroup$ – GiorgioP Dec 17 '19 at 21:14

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