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I. Introduction
1 Scope and Nature of Classical Physics
The ultimate aim of classical mechanics is to predict the trajectories of objects given that the nature of the interactions between them and their environments is known and that the initial states of the objects are provided. In introductory and intermediate-level physics courses, one often focuses on one or two objects and attempts to find their positions as functions of time by using Newton’s laws.
Example: Consider a block with known mass that is free to move across a frictionless horizontal surface. The block is connected to one end of a spring of known stiffness and the other end of the spring is attached to an immovable wall.
x
Figure I.1.1: Spring and mass system on a horizontal surface viewed from above. The dashed line indicates the equilibrium position of the center of the block.
Suppose that the mass of the block is m, the spring constant is k and x denotes the displacement of the center of the block from equilibrium. Newton’s second law applied to an object that moves in one dimension gives
Fnet = ma (I.1.1)
which gives −kx = d^2 x dt^2
(I.1.2)
where t is time. Via this route, the problem of determining the trajectory (i.e. the dependence of x on t) of a physical system has been reduced to the math- ematical problem of solving a second order differential equation. Fortunately the type of differential equation that appears in Eq. (I.1.2) can be solved fairly easily. The general solution is
x(t) = x(0) cos ωt + v(0) ω
sin ωt (I.1.3)
where ω =
k m
(I.1.4)
and the two constants x(0) and v(0) have the physical interpretation of the initial position and velocity respectively. The graph of position as a function of time is sometimes called a trajectory. Fig. I.1.2 illustrates an example of this for the block connected to a spring. x
t
Figure I.1.2: Position vs. time graph for a block connected to a spring.
With a little insight it is clear that such a plot of position vs. time indicates oscillatory motion and, in this sense, the trajectory offers a fairly concrete description of the block’s motion.
In the example of the spring and block system, we see that Newton’s laws eventually give an expression for the trajectory of the block. All other physically interesting quantities, such as velocity, momentum and energy, that pertain to the block can be derived from the equation for its trajectory.
Exercise: Show that the velocity of the block, as a function of time, is given by
v(t) = −ωx(0) sin ωt + v(0) cos ωt
and the total mechanical energy is given by
E =
mω^2 [x(0)]^2 +
m[v(0)]^2.
and it is easily seen that, in this case the x component of the force is
Fx = −kx. (I.2.7)
This is mathematically identical to the situation of the block and spring and is merely another example of a simple harmonic oscillator. In classical mechanics, the task would be to determine the trajectory of the ion. In quantum mechanics, it will emerge that there is no sense in trying to do this. In fact, the concept of force is inapplicable and so is Eq. (I.2.6) and everything that follows from it. The key mathematical entity in the quantum mechanical description of this type of particle is the wavefunction, Ψ(x, t), which can produce complex values. One example of a wavefunction for the ion in the above potential, or any harmonic oscillator, turns out to be as illustrated in Fig. I.2.3.
Ψ(x, 0)
x
Figure I.2.3: One possible wavefunction for a harmonic oscillator at t = 0.
The means by which one can arrive at a sensible wavefunction for a simple harmonic oscillator are typically described in a sophomore-level modern physics course. What is important to note at this stage, is that the wavefunction does not represent a trajectory of a particle as time passes. In fact the wavefunction as illustrated in Fig. I.2.3 is that at one particular instant. It therefore cannot resemble any kind of evolution of the ion’s physical state. What then, is its use in physical terms?
In general the state of a particle that is free to move in one dimension is described in quantum theory by a wavefunction Ψ(x, t). This is useful for determining the outcomes and the associated statistics of measurements. For example,
Prob(position measurement gives outcome from x 0 to x 0 + dx) = |Ψ(x 0 , t)|^2 dx (I.2.8)
and there are a variety of mathematical rules for predicting the outcomes of measure- ments of other physical quantities statistically. This is really the only interpretation of the wavefunction. Quantum theory provides the mathematical rules which enable one to de- termine, in such cases, suitable wavefunctions and to use these to determine the outcomes
of measurements. The wavefunction is then more of a mathematical tool than a means for visualizing the physical evolution that the trajectory offers. In this sense quantum theory is more abstract than classical physics. Quantum theory has been enormously successful in describing the physical world, partic- ularly at small scales. Quantum theory features prominently in the description of, amongst others,
- atoms and molecules and their interactions with light,
- subatomic particles (high-energy physics, particle physics),
- light at the low photon levels (quantum optics),
- the properties of solids, including thermal and electrical properties (solid state physics),
- superconductivity, superfluidity, and
- quantum information.
where r is the vector from the origin to the particle’s location and p is its linear momentum. For a particle undergoing uniform circular motion with speed v, the magnitude of the angular momentum is L = mvr (II.1.1)
where r is the radius of orbit. The direction of L is along the axis of rotation. The total angular momentum of the entire solid object is obtained by adding or integrating over the angular momenta of its constituent particles.
Exercise: Show that Eq. (II.1.1) is valid.
One can envisage measuring the orbital angular momentum of an extended object by determining each of the quantities on the right hand side of Eq. (II.1.1) and calculating L accordingly. However, if the particle is charged there is an indirect method for determining L which is suitable for objects on the molecular or smaller scales. This uses the facts that a moving charged particle, in effect creates an electric current and this current produces a magnetic field. For particles which move in circular orbits, the magnetic field produced by the particle and the interactions with external magnetic fields are predominantly described in terms of the particle’s magnetic dipole moment. For a loop of current of magnitude I that lies in a plane, the magnitude of the magnetic dipole moment is
μ = IA (II.1.2)
where A is the area of the loop. This is converted into a vector, μ, by stipulating that the direction of the magnetic dipole moment is perpendicular to the loop and oriented using the right hand rule. This is illustrated in Fig. II.1.2.
μ
I
Figure II.1.2: Orbiting loop of positive charge and the current loop produced. The area of the shaded loop is A. The direction of the angular momentum is indicated via μ.
A single charged particle, with charge q, which orbits at a constant rate, effectively establishes a current of magnitude
I = |q| T where T is the period of orbit.
Exercise: Show that the current produced a particle with charge q and moving in a circle of radius r and at speed v has magnitude
I =
|q|v 2 πr and show that this gives a magnetic dipole moment of magnitude
μ =
|q|vr 2
. (II.1.3)
Eqs. (II.1.3) and (II.1.1) yield μ = |q| 2 m
L
and taking directions into account for both positive and negative charges gives
μ = q 2 m
L
and for an extended object, whose mass density is proportional to its charge density, su- perpositions of each side give
μ =
Q
2 M
L (II.1.4)
where Q is the total charge and M the total mass of the object. For a rigid rotating object the total angular momentum is called the spin angular momentum and is denoted S. If the mass and charge distributions are not proportional (e.g. the mass is uniformly distributed throughout the object but the charge just resides on the surface), the relationship between the magnetic dipole moment and the spin angular momentum becomes
μ = g
Q
2 M
S (II.1.5)
where g is a dimensionless constant called the gyromagnetic or g-factor. This depends on the relative arrangement of the charge and mass distributions and its value can thus give some insight into the structure of the solid object. The same relationship extends to individual charge particles and with the mass and charge represented by m and q respectively, giving
μ = g q 2 m
S.
The interaction between a magnetic dipole and an external magnetic field, B, is deter- mined via the potential energy of the dipole in the field,
V = −μ · B (II.1.6)
B
Detector Screen
v
d
z
x
Figure II.2.3: Stern-Gerlach experimental setup. The two magnets produce an inhomogeneous magnetic field with constant direction in the gap between them and which is approximately zero beyond the gap. Particles fired into the gap with identical magnetic dipole moments are deflected by the same amount d.
represented by Sz (rather than d). It is typical to consider this entire experiment as nothing other than a measurement of Sz and the Stern-Gerlach apparatus as a merely a device for measuring Sz. Typically the beam of particles which is fired through the gap is prepared in such a way that there is no inherently preferred direction for each particle’s angular momentum prior to entering the magnetic field. This would imply that for classical particles the spin angular momenta, S, are randomly oriented with all directions equally likely. For classical particles, it follows that the continuous range of values of Sz in the range |S| 6 Sz 6 |S| are possible. Note that each particle will only arrive at one location.
B
Detector Screen
v
region in which particles arrive
Figure II.2.4: Stern-Gerlach experimental setup with randomly prepared classical particles.
The Stern-Gerlach experiment indicates that, as illustrated in Fig. II.2.5, for spin-1/ 2 particles such as the electron, only two deflections are possible, corresponding to Sz = +ℏ/ 2 and Sz = −ℏ/ 2. For randomly prepared particles, each only emerges at one location and approximately half emerging in the upper beam, for which Sz = +ℏ/2, and half in the lower beam, for which Sz = −ℏ/2. Experiments indicate it is impossible to predict at which location any of these randomly prepared particles will arrive.
Detector Screen
Sz = +
Sz = −
Figure II.2.5: Stern-Gerlach experiment for particles with randomly oriented magnetic dipole moments
The entire apparatus could be rotated through 90◦^ about the x axis so that the magnetic field is now oriented along the y-axis. It follows that this would measure the y-component of the particle’s spin, Sy. The possible outcomes of the experiment cannot be changed by a mere rotation of the apparatus and, for a spin-1/2 particle, must be Sy = +ℏ/2 or Sy = −ℏ/ 2. One can now conceive of a Stern-Gerlach experiment in which the magnetic field is oriented along an arbitrary direction, nˆ, in which case the Stern-Gerlach device is an apparatus for measuring, Sn, the component of particle spin along ˆn. Again the two possible outcomes are Sn = +ℏ/2 or Sn = −ℏ/ 2.
2.2 Schematic diagrams for the Stern-Gerlach experiment.
A schematic description of any of these Stern-Gerlach measurements requires a specification of the direction of the magnetic field, ˆn and the two possible locations in which the particle emerges, corresponding to Sn = +ℏ/2 or Sn = −ℏ/ 2. The entire arrangements of magnets can be represented by a box, labeled with ˆn and from which emerge two particle trajectories, each corresponding to one of the two outcomes. This is illustrated in Fig. II.2.6 and is called an SG nˆ measurement.
SG nˆ
Sn = +ℏ/ 2
Sn = −ℏ/ 2
Figure II.2.6: Schematic diagram of an SG nˆ measurement. The horizontal line on the left indicates the trajectory of particles fired into the apparatus. Those on the right are the trajectories corresponding to each of the two outcomes.
Definite states of physical systems such as those described above are denoted by a “ket”, an arrangement of brackets containing a label which indicates a certain outcome upon a particular measurement. These have the form
|label〉
where the contents of the ket are a label which is usually an abbreviated way of describing the particle’s physical state. From the description of sequences of SG zˆ measurements, we use
|+zˆ〉 ⇐⇒ measuring Sz yields Sz = +ℏ/2 with certainty, and |−zˆ〉 ⇐⇒ measuring Sz yields Sz = −ℏ/2 with certainty.
(II.2.3)
Thus the particle which emerges from the first apparatus in the upper trajectory is in state |+zˆ〉. However, neither of these states necessarily applies to particles entering the first apparatus, since these can emerge with either outcome for Sz. Note that the symbol within the brackets (here +zˆ or −zˆ) is a label which indicates the relevant measurement and outcome. This can be generalized to situations involving successive SG nˆ apparati, for some fixed direction nˆ. It follows that the corresponding states of the spin-1/2 particle are:
|+nˆ〉 ⇐⇒ measuring Sn yields Sn = +ℏ/2 with certainty |−nˆ〉 ⇐⇒ measuring Sn yields Sn = −ℏ/2 with certainty.
(II.2.4)
Schematically the physical meaning of the general states |+ˆn〉 and |−nˆ〉 phrased in terms of future measurement outcomes is provided in Fig. II.2.8.
|+ˆn〉 (^) SG nˆ
Sn = +ℏ/ 2 certainty!
Sn = −ℏ/ 2
never!
Figure II.2.8: The meaning of the state |+ˆn〉 in terms of an input to SG nˆ. The measurement outcome Sn = +ℏ/2 will occur with certainty.
A consequence of the relationships of Eq. (II.2.4) is that if a particle is subjected to an SG nˆ measurement and emerges with Sn = +ℏ/2 then it’s state immediately after measure- ment is |+nˆ〉. Similarly if it emerges immediately after measurement with Sn = −ℏ/2 then it’s state immediately after measurement is |−ˆn〉. An example is illustrated in Fig. II.2.9.
Example: Describe an experiment which is such that it prepares particles with cer- tainty in the state |+xˆ〉.
SG nˆ
Sn = +ℏ/ 2 |+ˆn〉
Sn = −ℏ/ 2 Figure II.2.9: The meaning of the state |+nˆ〉 in terms of an outcome after SG ˆn. A particle is supplied and if the measurement outcome is Sn = +ℏ/ 2 then subsequently the state is |+ˆn〉.
Answer: Suppose that you had particles which are, in some sense, randomly prepared. Each is subjected to an SG xˆ apparatus. Those that emerge with Sx = +ℏ/2 are in the state |+xˆ〉 and those that emerge with Sx = −ℏ/2 are in the state |−xˆ〉. Keep all of the former and discard any of the latter. Note that this will not work if, by some chance all the particles are in the state |−ˆx〉 prior to entering the SG xˆ apparatus. In this case more manipulations, to be explained later, are needed.
Thus there are infinitely many states available to a spin-1/2 particle corresponding to all possible directions nˆ. In fact, it will emerge that every state available to a spin-1/2 is of the form |+ˆn〉 for some unit vector nˆ. Physically this means that for any particle there is some (possibly unknown to the experimentalist) direction nˆ such that if the particle is subject to an SG nˆ apparatus the measurement outcome will be Sn = +ℏ/ 2 with certainty.
2.3.2 Repeated measurements of the different types
To understand how the many possible states are related to each other consider successive Stern-Gerlach measurements of different types. For example suppose that the particle is first subjected to an SG zˆ apparatus and then to an SG xˆ apparatus as illustrated in Fig. II.2.10.
SG zˆ
Sz = +ℏ/ 2
Sz = −ℏ/ 2
SG ˆx
Sx = +ℏ/ 2
Sx = −ℏ/ 2
SG ˆx
Sx = +ℏ/ 2
Sx = −ℏ/ 2
Figure II.2.10: Successive Stern-Gerlach measurements of different types.
Experiments show that if the particle emerges from the SG ˆz apparatus in the upper
SG ˆz
Sz = +ℏ/ 2
Sz = −ℏ/ 2
SG xˆ
Sx = +ℏ/ 2
Sx = −ℏ/ 2
SG zˆ
Sz = +ℏ/ 2
Sz = −ℏ/ 2
SG zˆ
Sz = +ℏ/ 2
Sz = −ℏ/ 2
Figure II.2.11: Three successive Stern-Gerlach measurements for the cir- cumstance in which the particle emerges with Sz = +ℏ/2 after the first SG ˆz measurement.
An unusual feature of quantum mechanics is evident when one considers two Stern- Gerlach measurements of the same type interspersed with a single Stern-Gerlach measure- ment along an orthogonal direction. An example is illustrated in Fig. II.2.11, in which it is assumed that the particle emerged with Sz = +ℏ/2 after the first SG zˆ measurement. Applying the rules of (II.2.6) one can deduce that the probability with which the parti- cle emerges with Sz = +ℏ/2 after either of the latter SG zˆ measurements is 1/ 2. Similarly the probability with which the particle emerges with Sz = −ℏ/2 after either of the latter SG zˆ measurements is 1/ 2.
Exercise: Consider the arrangement of Fig. II.2.11. Suppose that the particle emerges from the SG ˆx measurement with Sx = +ℏ/ 2. Show that the probability with which the particle emerges from the subsequent SG zˆ measurement with Sz = +ℏ/2 is 1/ 2. Repeat this for a particle that emerges with Sz = −ℏ/2 is 1/ 2. Repeat this entire analysis for the case where the particle emerges from the SG xˆ measurement with Sx = −ℏ/ 2.
Exercise: Repeat the argument of the previous exercise for the case where the particle emerges with Sz = +ℏ/2 after the first SG zˆ measurement.
The implication is that it makes sense to speak of a particle as having a definite value for Sz in the context where the only subsequent operations are SG zˆ measurements. However whenever subsequent operations include SG nˆ measurements where nˆ is distinct from zˆ (or −zˆ), it is impossible to speak of a particle as having a definite value of Sz. Thus is quantum mechanics it is meaningful to speak of a particle as having a definite value for Sn only in certain very restricted contexts. One cannot universally ascribe a definite value of Sn to a particle.
2.4 Maximal description of physical states
It is possible to consider measuring more than one component of spin via successive Stern- Gerlach apparati as illustrated in Fig. II.2.12.
SG xˆ
Sx = +ℏ/ 2
Sx = −ℏ/ 2
SG yˆ
Sy = +ℏ/ 2
Sy = −ℏ/ 2
SG yˆ
Sy = +ℏ/ 2
Sy = −ℏ/ 2
Figure II.2.12: Successive Stern-Gerlach measurements in an attempt to measure orthogonal spin components.
Any spin-1/2 particle to which these are applied will yield values for Sx and Sy. But this joint outcome can only be considered a measurement of a property of the particle if, when repeated again to a system that has undergone no extra measurements or interactions, it yields exactly the same value as it did previously.
Exercise: Consider the double SG experiment as illustrated in Fig. II.2.12 and a spin-1/2 particle that emerges from the uppermost output beam. Suppose that this is then reapplied to the same apparatus. Show that it does not emerge from the uppermost beam with certainty.
The preceding exercise indicates that two SG apparati cannot be combined to jointly measure two orthogonal components of spin in the sense that the measurement outcome is not repeatable. Thus, in the context of SG measurements it is not sensible to label the physical state of a particle via |+ mˆ, +nˆ〉 where mˆ and mˆ are not parallel; such a state would indicate that a combined SG mˆ and SG ˆn measurement would yield Sm = +ℏ/2 and Sn = +ℏ/2 with certainty. However, a general argument similar to that of the preceding exercise indicates that this is not possible. Thus, in terms of repeatable measurement outcomes, the only physically meaningful states are |+nˆ〉 and |−nˆ〉 where nˆ is any unit vector. The collection can be simplified by noting that, in terms of SG nˆ measurement outcomes, |−zˆ〉 is equivalent to |+(−nˆ)〉.
Exercise: Show that |+(−ˆn)〉 yields Sn = −ℏ/2 with certainty. To do so define mˆ = −nˆ and determine the probabilities with which |+ mˆ〉 yields Sn = ±ℏ/2.
Thus the collection of all possible physically distinct states of spin-1/2 particles is {|+nˆ〉} where nˆ ranges through all unit vectors.
It is important to note that the mean is an idealized quantity and that a given run of measurements will not necessarily yield the mean when averaged. Suppose that the outcomes of an experiment involving N measurements are s 1 , s 2 , s 3 ,... sN. This constitutes a sample of the probability distribution and the sample average is defined as:
m :=
N
∑^ N
i=
si. (II.2.11)
Typically for large N one expects that m ≈ 〈m〉.
Example: For an unbiased die, in one trial consisting of ten rolls, the following out- comes occurred: 2, 4 , 4 , 3 , 3 , 2 , 1 , 4 , 4 , 6. The sample average of these is 3. 3 which differs from the mean, 3. 5.
Various theorems in probability theory quantify the extent to which a sample average approximates the mean. Generally as N increases the probability with which the sample average will be within a given range of the mean increases; the probability with which it is beyond the a certain range of the mean diminishes as 1/N. An important tool in quantifying such discrepancies and fluctuations away from the mean is the standard deviation or variance of a probability distribution. This is defined as:
∆m :=
√√ ∑n
i=
pi (mi − 〈m〉)^2 (II.2.12)
and this quantifies the extent to which measurement outcomes typically deviate from the mean. A general result that simplifies this calculation is:
∆m :=
〈m^2 〉 − 〈m〉^2 (II.2.13)
where 〈 m^2
∑^ n
i=
m^2 i pi.
These notions can be applied to physical systems subject to the laws of quantum me- chanics.
Example: An ensemble of spin-1/2 particles is subjected to an SG mˆ measurement where mˆ = √^12 xˆ+ √^12 yˆ. Only the particles that emerge from this measure- ment having given the outcome Sm = +ℏ/2 are retained. Each of these is subjected to an SG xˆ measurement. Determine 〈Sx〉 and ∆Sx.
Answer: The particles that emerge from the initial SG mˆ measurement with Sm = +ℏ/2 are in the state |+ mˆ〉. We can list the measurement outcomes and probabilities for the SG ˆx measurement performed on particles in this state:
Outcome (Sz ) Probability
m 1 = +
p 1 =
m 2 = −
p 2 =
where we have used
Pr (Sx = +ℏ/2) =
(1 + mˆ · ˆn)
Pr (Sx = −ℏ/2) =
(1 − mˆ · ˆn)
etc,.... Thus
〈Sz 〉 =
∑^ n
i=
mipi
Then ∆Sz =
〈S^2 z 〉 − 〈Sz 〉^2. Here 〈 S z^2
∑^ n
i=
m^2 i pi
ℏ^2
Thus ∆Sz =
ℏ^2
ℏ^2
