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”Postulates of Quantum Mechanics” Are Often Stated in Textbooks. There

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Notes on the Formalism of Quantum Mechanics The ”postulates of quantum mechanics” are often stated in textbooks. There are two main properties of ”physics” upon which these postulates are based: 1)the prob- ability of measurements and 2) the interference properties of the probability amplitudes. These notes are meant to help students understand the ”physics” un- derlying the formalism of quantum mechanics. Throughout the discussion let A and B be two quantities that can be measured (observables). For simplicity, assume that each measurement yields values that are discrete, quantized. We will generalize later to measurements that yield continuous values. Label the possible values of observable A as αi and, and of observable B as βi. The integer i can take on a finite number or an infinite number of values. Probability of Measurements Experiments indicate that one can not determine the exact outcome of a measure- ment before the measurement is carried out. The only information about the outcome of a measurement are the probabilities for the various outcomes. If a measurement of A is guaranteed to yield the result αi, then we will call this state a ”proper state” of the observable A and label it as α >. If a measurement of A is carried out on a | i system in the state αi >, then the outcome is always αi. However, a system is not always in a proper state| that is guaranteed to give a particular result. In general, a system will be in a state that has a certain ”probability distribution” to be in the var- ious αi states. One can represent the state Ψ > of a system as a linear combination of the α > states: | | i Ψ >= a1 α1 > +a2 α2 > +... = ai αi > (1) | | | X | What is the meaning of the ai’? One might be lead to believe that the ai represent the probability that a measurement of A yields a result αi. However, experiments also indicate that interference effects occur with the probabilities. Thus, in order to have the probabilities include interference effects we are lead to interpret the ai as a probability amplitude: The probability for obtaining a value of αi if the observable A is measured is a 2. | i| Similar reasoning and notation can be used with any other observable. For exam- ple, for the observable B, we have Ψ >= b1 β1 > +b2 β2 > +... = bi βi > (2) | | | X | 1 where Ψ > represents the state of the system, and b 2 is the probability that a | | i| measurement of B will yield the value βi. The above expansion is a result of the probabilistic nature of the outcome of a measurement. An important quantity in quantum mechanics is the probability to obtain a value of βi if B is measured when the system is in the state αj >. We will label this 2 | probability as < βi αj > . We will discuss why this method of labeling is valuable in a moment.| The quantity| | < β α > is a probability amplitude, whose square is i| j a probability. In terms of experiments, this would mean that the observable A is measured with a result αj >. Then observable B is immediately measured. The probability of obtaining β| when B is measured is < β α > 2 i | i| j | Interference: Superposition of probability amplitudes An unusual property of systems where quantum effects are important is that probability amplitudes add, which leads to interference effects of probabilities. These interference effects are demonstrated in many experiments. In class for example, we see these effects from single and double slit scattering experiments with electrons and photons. In lecture we let photons pass through two slits in a metal foil and land on a screen behind the slits. The alternating light and dark pattern on the screen can be understood from the superposition of probability amplitudes. The probability amplitude to for the photon to land at a certain point on the screen is the sum of the probability amplitude for the photon to pass through the right slit plus the probability amplitude for the photon to pass through the left slit. The probability itself is the absolute square of the total probability amplitude. For the double slip experiment the probability ampliude for passing though each slit is proportional to eipd/h¯ , where p is the momentum of the photon and d is the total distance traveled from source to screen. The total probability amplitude is: P robamp eipd1/h¯ + eipd2/h¯ (3) ∝ The probability (or probability density) is just the absolute square of the probability amplitude or: P rob eipd1/h¯ + eipd2/h¯ 2 (4) ∝ | | p(d2 d1) 2 + 2 cos( − ) (5) ∝ h¯ p∆d cos2( ) (6) ∝ h¯ 2 π∆d P rob cos2( ) (7) ∝ λ where λ is the DeBroglie wave length of the photon or electron, and ∆d = d2 d1 is the difference in the distance the particle takes going through right versus the− left slit. There is no chance for the particle to land at points on the screen where the path difference ∆d is half a wavelength. Stated in words, the linear superposition of probability amplitudes means (where prob means probability, and probamp means probability amplitude): Prob(for the system to be measured in the state β >) is equal to probamp | i | (system is in state α1 >) < βi α1 > + probamp(system is in state α2 >) < βi α2 > + .... + probamp|(system is in| state α >) < β α > + .... 2 | | | j i| j | or expressed in summation notation: prob(system to be measured in the state βi >) equals sum over j of probamp(system is in state α >) < β α > 2 | | | j i| j | Linear Vector Space of the States of a System In order for a set of objects to be a vector space, any linear combination of elements must also be an element of the vector space. Consider the set of all possible states of a system. For this ”state space” to be a vector space, any linear combination of states of a system must also be a possible state of the system. If we consider as the ”basis” elements of the system space all the possible outcomes of a measurement, say measurment A, then Ψ >= ai αi > is a possible state of the system for all | 2 P | possible ai as long as ai = 1. The field of the vector space is the type of quantity P | | that the ai are. For the state space of quantum systems, the ai must be complex (at least). Since any linear combination of states is also a state of the system, the set of all states of a system form a vector space over the field complex numbers. For a set of objects to be an inner-product vector space, a scalar product must be defined. The scalar product must be an element of the field (i.e. a complex number). We label the scalar product between state Ψ1 > and Ψ2 > as < Ψ2 Ψ1 >. The scalar product must have the following ”linear” properties:| | | < Ψ ( Ψ1 > + Ψ2 >)=< Ψ Ψ1 > + < Ψ Ψ2 > (8) | | | | | < Ψ (c Ψ1 >)= c< Ψ Ψ1 > (9) | | | 3 and ∗ < Ψ2 Ψ1 >=< Ψ1 Ψ2 > (10) | | The first two properties are true if: the interpretation of the scalar product is a probability amplitude, and probability amplitudes obey the linear super- position principle. The last property is because the field is the complex numbers. Thus, because of the linear superposition property of probability amplitudes, the set of all possible states of a system form a linear vector space with the scalar product interpreted as a probability amplitude. Basis sets of the Vector Space A basis for a vector space is a set of elements such that any state can be expressed as a linear combination of the basis elements. Since the vector space elements are states of a physical system, it is natural to use as a basis the states of an observable corresponding to a measurement outcome. For example, one could use the states of the observable A, αi >, as a basis. This means that every state of the system can be expressed as combinations| of states for all possible values of measurement A: Ψ >= ai αi > (11) | X | where the sum goes over all possible outcomes of A. This is a complete basis when it comes to measurements of A, since very possible outcome is included. However, this basis might be incomplete when it comes to all possible measurements and outcomes. Our physical interpretation of the scalar product allows a nice interpretation of the a ’s. The scalar product < α α > is the probability amplitude that a measurement i i| j of A will result in a value αj when the system is in the state αi >. The probability must be zero unless i = j. If i = j, the probability is 1. Thus, we have: < α α >= δ (12) i| j ij In the terms of linear algebra, the basis αi > is an orthonormal basis. If we take the scalar product of the previous equation| with α >, one obtains | j < αj Ψ >= ai < αj αi >= aj (13) | X | Thus, aj is interpreted as the probability amplitude that the state Ψ > will be found in the state α >, and a 2 is the probability that a measurement will| yield the value j | j| αj. 4 One could use the proper states of any observable as a basis. For example, one could use the proper states of the observable B: β > which correspond to values of | i βi if B is measured. Ψ >= bi βi > (14) | X | For these basis states the following is also true: < βj βi >= δij, bj =< βj Ψ >, with b 2 being the probability that a measurement B on the| system yields β .| | j| j If one knows all the probability amplitudes < βi αj > for all i and j, then all measurement probabilities of the two observables A and| B can be determined.
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