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Mathematical Minimum (for Aspiring Physicists)


Mathematical Schrodinger's Cat (QM Basic)

The followings are mostly about the application of quantum mechanics to specific cases. See "Short-cut to the Introduction of Quantum Theory" for a brief review of its theoretical base.

Square Well Wave Function Figure 05a (A) shows an elastic ball moves in an one dimensional box bouncing back and forth between the wall as described by classical mechanics. Its motion can be represented by x = vt, where x is the coordinate between the wall from 0 to a (the size of the box), t is the time and v is the velocity which switches sign whenever hitting the wall. In quantum mechanics for a small particle in such a square well potential, the dynamic is prescribed by the Schrodinger's equation :
      

Figure 05a Square Well Wave Function [view large image]

 Square Well Potential           Schrodinger's Equation

The solution, which satisfies the boundary conditions (0) = (a) = 0, is :



Figure 05a (B, C, D) shows the stationary wave functions with n = 1, 2, 3 respectively (blue for the real part and red for imaginary part from the time dependent wave function e-int = cos(nt) - i sin(nt)). The corresponding energy a.k.a. eigenvalue (eigen means special in German. In this case, it is special in the sense that the operation returns the same operand multiplied by the eigenvalue) En = n, and normalization constant B = (2/a)1/2. Note that the probability (always a real number) of finding the particle within a small segment x is Pn(x) = n*(x)n(x) x = (2x/a) sin2(knx). It does not depend on time - that's why it is called a stationary state. While in classical mechanics Pclassical = (x/a), it does not depend on space as well. Note also that both cases tend to zero as a , i.e., when the box is very large - it becomes very difficult to find the particle anywhere.

As mentioned in the last section, the linear combination of the stationary wave functions n is also a solution (albeit not stationary and no definite eigenvalue). It can be expressed in the general form :
.
The coefficient cn is interpreted as the probability amplitude of the nth state in the combination (superposition). It can be determined by the initial condition at t = 0 with any regular function (not necessarily a solution of the Schrodinger's equation) such as :
,
where the normalization constant A = 4/(5a)1/2.
Multiplying the superposition equation at t = 0 by sin(knx) and applying the orthonormal relation yield
cn = A B ,
which can be evaluated further by using the identity Sin3(y) = (3/4) sin(y) - (1/4) sin(3y) : c1 = 3/(10)1/2, c3 = -1/(10)1/2, all the other cn = 0.
Probability of Stationary States Probability of Superposition state It follows that the probability of the particle in n = 1 state is |c1|2 = 0.9, and |c3|2 = 0.1 for n = 3. Figure 05a (E, F) show the behavior of the non-stationary wave function. The probability of finding the particle at x now depends on both space and time (Figure 05c) :

Figure 05b shows 3 of the stationary states (n = 1, 2, 3), which are independent of time. In contrast, the superposition state varies over time. The time variation is dispicted in color running from = 0 (black curve) to (yellow curve), and then back to 2 as indicated by the labels beside the curves [ = (3 - 1)t].

Figure 05b Probability of Stationary States [view large image]

Figure 05c Probability of Superposition State [view large image]

Note that in the superposition state E = (3 - 1), t = 2/(3 - 1) given Et = h. Not surprisingly, it just runs a circle and returns to the very first principle of quantum theory - the uncertainty principle.

Schrodinger's Cat In Schrodinger's Cat Paradox, the cat is to assume two states - alive and death. We can identify the wave functions 1 and 3 to these two states respectively. If we look at the probability P(x,t) closely, it is the time dependent interference term that mixes up the two states. However, if we include the environment such as the box (enclosing the cat) into the system, these two states would entangle with billions particles producing phase difference randomly. Thus, the cross term would vanish by averaging over all the entanglements. The effect is called de-coherence, which happens in very short time interval (~ 10-27 sec) in general. The system is said to be "collapsed" into a definite eigenstate - 1 or 3 in this mathematical analogy.

Figure 05d Schrodinger's Cat [view large image]


A very small disturbance may not be sufficient to de-cohere the system as shown in the following example. Supposed a small perturbation eu is introduced into the infinite square well from time 0 to t, where e is a coupling constant and u could be a function of x and t, the probability amplitude cn would then be a function of t. It can be shown that :



If a constant but small electric E is applied to the infinite square well with a charged particle, then
u = Ex, u11 = u33 = (a/2)E, and u13 = u31 = 0. It follows that :

c1(t) = c1(0) exp[-i(e/)u11t],
c3(t) = c3(0) exp[-i(e/)u33t],
which shows that for this case the modification just adds the small perturbation energy to the un-perturbed energy En, there is no transition crossing the stationary states since unm = 0 for n m. For the effect of a full-blown static electric field (not just a small perturbation), see "Images of Hydrogen Atom".

The other solvable case is to introduce a weak oscillating electric field in resonance with the stationary state wave function, e.g., u = Ea0sin(k1x). The matrix elements work out to be : u11 = 0.849 Ea0, u33 = 0.652 Ea0, and u13 = u31 = - 0.173 Ea0. In order to obtain a more legible form, a further assumption is made on the "turn-on" time such that t << /(eu11). The time variation of the probability amplitudes then take the form :

The first term is the usual addition of the perturbation energy (albeit in different amount). It is the second term which induces transition between stationary states. However, the movement in this case is only partial (via oscillatory change of the probability amplitudes); a complete transition would reduce one of the amplitudes to 0. Incidentally, if the oscillating electric field is in anti-resonance to the stationary state wave function, e.g., u = Ea0cos(k1x), then all the unm vanish leaving all the cn unchanged.

Quantum Measurement The experiment using flux qubit is more instructive to show pictorially the transition in quantum measurement process. Figure 05e shows that the superposition of the spin up and down states (cat alive and death) can point the spin axis to any direction depending on the amplitudes a and b. The measurement (acts as the environment) "collapses" the superposition to a definite spin state either up or down. This example uses a SQUID detector ("Superconducting Quantum Interference Device" - a very sensitive magnetometer used to measure extremely subtle magnetic fields) to perform the measurement.

Figure 05e Quantum Measurement

(a) Measurement of superposition flux qubit state and state reduction.
(b) State detection by SQUID.

(see "bra-ket" notation for the weird symbol |A>)

Collapse of the Wave Function The "collapse" of the superposition state to a stationary state in the process of measurement is a controversial subject until recently in 2013 when it is demonstrated in slow motion to show the way the process unfolded (reported in Nature, 10 October 2013). The experimental observation is portrayed by an artist's analogical rendition (Figure 05f), in which the butterflies in a cage (in superposition state) make their ways erratically during the measurement toward either one of the two trees (representing the 2 stationary states). Each butterfly in the picture corresponds to one experimental observation.

Figure 05f Collapse of the Superposition Wave Function into Stationary State [view large image]



BTW, if instead of an infinite square well, the potential V becomes a barrier with width a and hight Vo, then part of the incident wave will be reflected while another part will penetrate the barrier and exits to the other side as transmitted wave (Figure 05g). This is called quantum
Quantum Tunnelling Standing Wave tunnelling, which is possible only through the treatment by quantum mechanics. A classical particle with energy E < Vo will just bounce back with no chance of going over the barrier to the other side. The phenomena of alpha decay, and other application can only be explained by such quantum effect. The amplitudes of the various waves are determined by the boundary conditions. With the additional assumption of E << Vo, the transmission coefficient T :

Figure 05g Quantum Tunnelling [view large image]


Figure 05h Standing Wave
[view large image]


T = |A|2/|D|2 = 16 [E/(Vo-E)] exp(-2ap2/),
where p2 = [2m(Vo-E)]1/2.

As Vo , T 0. If there is a similar wall on the left side, the barrier becomes an infinite well, and the combination of incident and reflected waves will become standing (stationary) wave (Figure 05h).

The mathematical formulation and terminology in the following are optional, but they would often creep up unsuspectedly in any QM course.


















The usage of bra-ket notation is especially convenient in the Quantum Field Theory, where the base vectors are not functions with continuous variables such as space and time. The wave functions or fields have been quantized to be associated with the (particle) number operator, which operate on the base vectors labeled by some parameters such as momentum, spin, and other quantum numbers.

See "The Different Perspectives of the 1st and 2nd Quantizations".

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