Notes for Class Meeting 19: Uncertainty

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Piers Simpson
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1 Notes for Class Meeting 19: Uncertainty Uncertainty in Momentum and Position In 1926, Werner Heisenberg formulated the uncertainty principle: It is impossible to determine both the momentum and position of an object to infinite precision. Explicitly, ΔxΔp x 2, (19.1) or in words, the uncertainty of the position of an object in some direction times the uncertainty in its momentum in that direction is always greater than or equal to / 2. A couple of points should be noted: (1) It is only a the product of the uncertainty of the position in some direction and the uncertainty in the momentum in that direction that enters the uncertainty principle. For example, there need not be any inherent uncertainty between the position in the x direction and the momentum in the y direction. (2) The value / 2 is the absolute minimum uncertainty and only occurs in one special case. More common values are or h/2, depending on the case, but the exact value will not be important for our purposes. In his lecture, Feynman explained that the uncertainty principle arises because any attempt to measure the position of a particle will disrupt its momentum. This is not the best explanation, but let s explore it first. Why is the two-hole experiment an example of the uncertainty principle? Let s solve for the first minimum in the two-hole interference pattern, which we will call x, the distance from the center maximum. This will occur when the distance from one hole is a half wavelength longer than the distance from the other hole, since this will cause the peak of a wave from the first hole to coincide with the trough of a wave from the other hole. Thus, the condition is L 2 L 1 = λ / 2. (19.2) From the Pythagorean theorem,

2 2 L x and L x. (19.3) After a bit of algebra, which we will consign to the appendix, we obtain x L = λ 2d. (19.4) The right hand side of Eq. 19.4, x/l is the change in angle, Δθ, between the center maximum and the first minimum of the interference pattern. Thus, the change in the x component of the electron momentum, p x, that would destroy the interference pattern is just Δ p x = p Δθ = h λ Δθ = h λ λ 2d = h 2d, (19.5) where we have used the de Broglie relationship that p = h / λ. Therefore, to avoid destroying the interference pattern, we must change the x component of the electron momentum by less than h/2d. But to detect which hole the electron went through, we need to use a photon with a wavelength less that the distance between the two holes, λ < d, (19.6) This is because, in some sense, the photon size is comparable to its wavelength. From the de Broglie relationship, the photon has a momentum p γ = h / λ. (19.7) Bouncing a photon of this momentum off of the electron would cause a change in the x component of the electron momentum by about Δp x 2 p γ = 2 h λ > 2 h d, (19.8) where the last piece of Eq comes from using Eq Thus, since the change in the x component of the electron s momentum, Eq. 19.8, is larger than that required to destroy the interference pattern, Eq. 19.5, the interference pattern will be destroyed by detecting which hole the electron went through. 1 The above argument, which Feynman made in his lecture, has two advantages: (1) A single counterexample can show that quantum mechanics is inconsistent (which is what Einstein tried to do in the Einstein-Bohr exchanges). (2) It uses classical arguments, so it is easy to understand. However, I stated above that this argument is not the best explanation for the uncertainty principle, and the reason is that it has two serious disadvantages: (1) You can never list all of the ways of detecting which hole the electron went through, so you cannot use it to establish a general principle. (2) More important, it encourages you to think that the electron has a position and momentum, but that you simply cannot measure it. This is incorrect. At any given time, the electron has neither a definite position nor a definite momentum. 2 1 A more precise analysis involves using a lens to focus the photon, but this is not important for our purposes. In fact, it is important for you to understand the physics argument that is being made, but not important for you to follow all of the mathematics.

3 The reason why Feynman is sure that the uncertainty principle will hold in every case, even though he cannot examine every case, is that the uncertainty principle is inherent in the mathematics of waves. There is a mathematical theorem known as the Fourier theorem, which states that any curve can be made from a (possibly infinite) sum of sine and cosine functions of different wave lengths. A single sine function, say sin(2π x / λ), has a single wavelength λ and thus corresponds, through the de Broglie relationship, to a single and absolutely precise momentum. However, it extends over all space. If you want to localize the wave, you have to combine several different wavelengths, and thus have some uncertainty in its momentum. The more the wave is localized in space, the more uncertain the momentum becomes. I will demonstrate this during the class meeting. We will discuss the concept of measurement in quantum mechanics in the next set of notes, but we will just note here that if one has a particle with a range of position and momentum then a precise measurement of either will change the wave to make the other more uncertain: What the precise value of x or p becomes is purely probabilistic. This is the part of quantum mechanics that, according to Feynman, nobody understands. The uncertainty principle guarantees that the future is unpredictable in principle, as well as in practice. If you know where a particle is precisely, then you do not where it is going. If you know where it is going (i.e., its velocity from its momentum), then you do not know where it started.

4 Why do we not see the consequences of the uncertainty principle in our everyday lives? Simply because h is extremely small, so the uncertainties are not noticeable in macroscopic objects. 2 Uncertainty in Energy and Time What does the uncertainty principle tell us about time in quantum mechanics? Consider a particle moving in the x direction. Energy and momentum are related, E = 1 2 mv2 = 1 2m (mv)2 = p2 2m, (19.9) so any uncertainty in momentum will also mean an uncertainty in energy. With a bit of calculus, one can show that ΔE = p mv Δp = m m Δp = vδp Δp = 1 ΔE. (19.10) v If we sit at one point in space and ask at what time does a particle pass us, then an uncertainty in x will imply an uncertainty in the time of passage: If time early is t = 0, then time late is t = x v. Therefore, Δt = 1 Δx Δx = vδt. Combining this with Eq , we obtain v 2 This is not completely true. A favorite problem for beginning quantum mechanics students is to ask them to calculate how long you can balance a perfectly sharpened pencil on its tip. The uncertainly principle will require the eraser end of the pencil to have some uncertainty in both its position and its momentum. This will cause it to fall over in less than four seconds.

5 Δp x Δx = 1 v ΔE v Δt ( ) = ΔE Δt 2 (19.11) What are we to make of Eq ? In standard works on quantum mechanics, you will find statements that time is just a parameter and cannot have any uncertainty. The ΔE Δt / 2 relationship is interpreted as follows: An energy can only be measured to a certain precision in a finite amount of time. 3 While I certainly agree with the above statement, I want to assert a more unconventional interpretation of ΔE Δt / 2, since I believe that you are already too sophisticated to accept the standard interpretation as the full interpretation. We have learned through relativity that space and time are closely related and form a fourdimensional spacetime. Therefore, if a particle s position is uncertain, then a particle s time must also be uncertain. Thus, it follows that time in quantum mechanics is fuzzy in the same way as space is fuzzy. Consequences of the Fuzziness of Time This concept can help us understand why energy states are quantized in the Bohr atom. Say we want to measure the energy of an atom to high precision. From the uncertainty principle, this will take some period of time. Consider two cases, one that meets the Bohr condition that each orbit contains an integer number of wavelengths and one that does not. If a measurement takes some period of time, it implies that we are averaging over that time. Assume that the Bohr condition is not met. Then on each revolution, the wave will have a different phase, and over a long period of time the peaks and troughs will all average to zero. However, if the Bohr condition is met, then all of the waves from successive revolutions will be in phase with peaks matching peaks and troughs matching troughs and we will have constructive interference. This picture then explains why only discrete orbits exist in the Bohr model. It also goes further in explaining why excited states, which decay by emitting a photon, do not have a completely precise energy, but in fact have some spread in energy given by the ΔE Δt / 2 uncertainty principle. If a state lives for a short time, then if the energy is not quite at the center value, it will not have time to get completely out of phase before it decays, since the peaks will move only a small amount on each successive revolution. To be concrete, assume that a state lasts for 100 revolutions, then the energy (which determines the wavelength) will have to agree with the Bohr condition to about 1 part in 100. If we measure the energy of the photon, it will vary from event to event by about 3 I draw your attention to the sourcebook reading on the Bohr-Einstein exchange in which Einstein tried to give a counterexample to this uncertainty principle. See also my interpretation of this in problem 226 in the sourcebook (p. 333, with the solution on p. 402).

6 1%. If there is only time for about one revolution, then any orbit is as good as another and there is no energy quantization. Comparison of the Position-Momentum and Time-Energy Uncertainty Principles Even though we derived the time-energy uncertainty principle from the positionmomentum uncertainty principle, they are quite different in general due to the vector character of momentum. For example, consider a hydrogen atom in its ground state. The position-momentum uncertainty relates the uncertainty in momentum, which is about twice its magnitude, since we do not know which direction it is in, to the diameter of the atom, since we do not know where the electron is in the atom. However, the time-energy uncertainty for the ground state is that the energy can be perfectly defined since the ground state can last forever. Quantum Tunneling Another consequence of the fuzziness of time, or if you prefer, the time-energy uncertainty principle, is the quantum mechanical property of tunneling. Consider a particle trapped in a potential well, for example a ball rolling back and forth on a track, as shown below on the left. If the dashed line shows the total kinetic and potential energy of the ball, then it can never escape, because it does not have enough energy. However, the time-energy uncertainty relation tells us that energy conservation can be violated, if it happens for a sufficiently short time. Thus, a quantum mechanical ball would have some probability of escaping the potential well and appearing on the other side, as shown on the right. Energy Energy Position Position Quantum tunneling is an important effect. It is responsible for particle decays, some effects in electronics, and the burning of hydrogen in the sun. We will demonstrate a form of quantum tunneling in the class meeting. Appendix: Algebra to Get from Eq and 19.3 to 19.4 To recap, the condition for destructive interference is L 2 L 1 = λ / 2. (19.2) From the Pythagorean theorem,

7 2 L x and L x. (19.3) Subtracting the two equations in Eq from each other, L 2 2 L 2 1 = L 2 d 2 2 x L 2 d 2 2 x (19.12) (L 2 L 1 )(L 2 L 1 ) = d 2 4 dx x2 d 2 4 dx x2 = 2dx. (19.13) Since L d, it is a good approximation to set (L 2 L 1 ) 2L in the left hand side of Eq We also use Eq for (L 2 L 2 ) to obtain λ 2L = 2dx 2 (19.14) x L = λ 2d. (19.4) 2

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