# What does "by an amount" mean here?

In A Very Short Introduction: Black Holes by Katherine Blundell, the author discusses the emptiness pf the space:

One of the surprising consequences of this new theory was that there were fleeting moments when it seems like energy needn’t be conserved. The first law of thermodynamics, the grand and seemingly unbreakable principle of physics, insisted that at every moment and at every place there had to be a strict accountancy between energy debits and energy credits. ‘Energy must always balance!’ thunders the Cosmic Accountant. In fact, it seems that the Universal accountancy rules are more lenient and it is possible to obtain credit. It is perfectly acceptable to borrow energy for a short period of time as long as you pay it back quickly afterwards. The amount you can borrow depends on the duration of the loan, by an amount described by the Heisenberg Uncertainty principle.

What's exactly meant by that? the degree of dependence?

• Single-letter typo: pf should be of. (Please feel free to flag this comment as obsolete once that's corrected.) Here's an edit link which should work for you, but not for me.
– TRiG
Commented Jun 30, 2022 at 12:12
• This quantum effect is of course precisely what allows a tunnelling diode to work. These are real devices in the real world that you can see in action, that "should not work" according to classical physics. Commented Jun 30, 2022 at 19:38

This is a picturesque description of the uncertainty principle which states that the product of the uncertainties of certain variables must always exceed a certain value. The wording "by an amount" refers to that value.

#### Conjugate variables

In quantum mechanics, such pairs of variable are known as "conjugate", or "complementary" variables. Arguably the most famous pair is momentum $$p$$ and position $$x$$. Denoting the uncertainties by $$\Delta$$, the principle is then $$\Delta_x \Delta_p \ge \hbar/2$$, where $$\hbar$$ is Planck's constant (divided by $$2\pi$$).

Other pairs include angular momentum & orientation, electrical potential & charge, and energy & time. The latter is what is being referred to in your example, and the text then says that

the amount $$\Delta_E$$ of energy you can burrow depends on the duration $$\Delta_t$$ of the loan, by an amount $$\hbar/2$$ described by the Heisenberg uncertainty principle.

#### Real-life example

An example of this energy-time variety is seen in the width of a spectral line: If an electron is excited to a higher energy from which it may de-excite by emitting a photon, that photon will have an energy equal to the energy difference between the two energy levels. The electron will not de-excite immediately, but will remain excited for a little time $$\Delta_t$$. This means that the energy difference is almost well-defined, but will have an intrinsic uncertainty $$\Delta_E \ge \hbar/2\Delta_t$$. For this reason, the emitted line is not an infinitely thin delta function, but will have a narrow, but finite, width.

However, unless the atoms emitting the photons lie very still (i.e. a very cold gas), the width will usually be dominated by the Doppler shifts caused by the thermal motion of the atoms, broadening the line by redshifting and blueshifting individual photons (at least in the center of the line).

The sentence is overly ornate; both uses of "amount" refer to the same quantity. A more concise - and less confusing - phrasing could be:

It is perfectly acceptable to borrow energy for a short period of time as long as you pay it back quickly afterwards. The amount and the duration are interlinked, in a relationship specified by the Heisenberg Uncertainty Principle.

The HUP sets a lower bound on the uncertainty, which itself is a relationship between a given probability and a corresponding upper bound on the product of "loan amount" and "loan duration".

• The last paragraph is not correct, since 1) it's a lower, not upper, bound, and 2) there's no probability of "not paying it back". The first part is correct, but I don't think it really says anything the other answer doesn't…
– pela
Commented Jul 1, 2022 at 6:10
• @pela The HUP provides a lower bound on the uncertainty, which is itself an upper bound on the "borrowing". (A lower bound on the borrowing would imply that in your example a spectral line would bifurcate.) There's a lower bound on how often the bailiff comes knocking. Commented Jul 1, 2022 at 7:52
• @pela while your answer is indeed more rigorous and precise, I've endeavoured to provide a simpler answer for people who are keen stargazers but not particle physicists. Commented Jul 1, 2022 at 7:59