Introduction to Quantum Mechanics I
Introduction to Quantum Mechanics I PHY 567
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This 6 page Class Notes was uploaded by Ms. Bryce Wisoky on Wednesday October 21, 2015. The Class Notes belongs to PHY 567 at Syracuse University taught by Staff in Fall. Since its upload, it has received 45 views. For similar materials see /class/225636/phy-567-syracuse-university in Physics 2 at Syracuse University.
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Date Created: 10/21/15
Lecture 7 Vectors Operators and the Hamiltonian evolution We now turn to a more abstract discussion of QM We have seen that in wave mechanics a quantum system is described mathematically by a func tion of space and time called the wavefunction This wavefunction forms a convenient representation for the more abstract notion of quantum state Remember that the description of a quantum state is quite different from its classical counterpart it must be consistent with the Heisenberg uncertainty principle and the principle of superposition Thus a quantum state describing a single particle may admit the possibility that a measurement of the par ticle s position or momentum may return more than one value To explain double slit interference we must also assume it may be split and recombined to yield other new quantum states Thus the mathematical quantity which is used to represent this quantum state should contain a large number of different component pieces of infor mation and two such objects may be added together with different weights to produce another such quantum state object If we look around in mathe matics for objects which behave in this we will see that quantum states have the same properties vectors Unlike the familiar vectors of three di mensional space the quantum state vectors inhabit a socalled complex vector space whose dimension may be infinite To see how this all works out lets summarize the important properties of ordinary vectors once this is done it will be obvious which of these properties carries over to more general vector spaces These properties are follows 1 The sum of two vectors a and b is a vector C a b 2 We can multiply any vector by a scalar to yield another vector eg b Aa C Vectors can be expanded in components that is it is possible to choose a suitable basis set of vectors like ijk of Cartesian coordinates and express every vector a sum over basis vectors weighted by real number coefficients a aii ajj akk The set of 3 numbers aig aj are then the components of the vector 4 There exists a dot product between two vectors a and b denoted ab which is just a scalar If a and b are expanded on a Cartesian basis this is just ab 115139 ljbj akbk The components of a vector are then nothing else than the dot product of the vector with the unit basis vectors eg ai ia The generalization that is needed to discuss vectors for QM is 0 Introduce a new notation a gt for quantum state vector Replace real scalars by complex scalars vector components are also in general complex numbers Let the dimension of the space in which QM takes place be big you like Thus the expansion of a state vector in components can be written gt3 Z Mild139 gt 1 where the set of vectors 33139 gt forms a basis in the space The exists a dot product which combines two vectors to yield a complex number This is denoted lt a b gt The analog of a Cartesian basis is one in which the basis vectors are arthaganal lt ci cj gt 615 The components m can just be interpreted the dot products of the basis vectors with the vector eg m lt cm gt The dot product between two vectors a gt and gb gt in such a basis is now ab afbi agbj 126 The latter result can be understood if we allow ourselves two types of vector the original sometimes called kct Vector a gt and a dual bra vector lt a whose components a with respect to the dual bra basis lt 6139 are just the complex conjugates of the ket components a if The vector lt a dual to a gt is sometimes also called the adjamt vector Notice that this de nition of the dot product ensures that the dotproduct of a vector with itself is a positive real number this will be a necessary ingredient in order to allow for the probability interpretation of the theory it will allow us to nor malize the quantum state vector to be of unit length lt a a gt 1 completely analogously to the normalization of the Schroedinger wavefunction With this technology we can expand the state vector 31 gt on some n dimensional basis 6139 gt we will assume from now on the basis is arthanar mal n i l gt EMF139 gt i1 The coefficients are just the generalized dot products of the state vec tor with the basis vectors u i lt 6131 gt Let us postpone discussion of what appropriate set of basis vectors to choose We will see that this is in timately connected to the choice of observable to measure Just notice that this mathematical expansion of the state vector embodies the physical prin ciple of superposition that is the quantum probability wave can be the sum of many contributions each of which may correspond to classically distinct possibilities for example each basis vector might represent a possible loca tion of the particle on the xaxis the quantum state is a sum over these allowing the particle to be simultaneously at many positions and forcing a probabilistic interpretation of the theory The evolution of the state vector follows from the Schroedinger equation of wave mechanics 8311 gt at The quantity H is called the Hamiltanian Operator Its purpose is to trans form one vector into another neighboring vector It has the dimensions of energy a primary observable Normalization of the wavefunction translates into the statement lt 11131 gt 1 ie the state vector is a kind of unit vector This normalization condi tion necessary for a probabilistic interpretation of the theory and it is necessary that it remain true for all time thus the state vector evolves in time it remains always of unit length What properties must H have in order that this be true Imagine solving the equation 1 over a small time period At m 11311 gt 1 It At gt 31 gt ltH lt gt To check the normalization condition we need to introduce the concept of adjamt operator Suppose gt gt then the adjoint vector lt d is given by lt lt B This is often written lt m which emphasizes that Bf operates on the bra vector here lt Thus we nd that iAt lt 111t At llt At gtlt magit gt 7 lt 1tHT HIlt gt Thus we require the Hamiltonian to be a self adjamt or Hermitian operator H H l We will see that hermitian operators play a central role in quantum mechanics Notice that our argument implies that the operator 1 applied to any vector preserves its length to 0At2 In the limit of vanishing At it is an example of a unitary operatmquot U Such operators U have the property Uf U l where U 1 is the inverse operator the operator which undoes the effect of U To see this is norm preserving consider gtU gt lt aUlU a gtlt a a gt In general such an operator may be written exp 1H where the operator H is hermitian not necessarily the Hamiltonian This is the case for the Schroedinger evolution eqn 1 which has a formal solution llt gt exp thh lU gt Notice that general unitary operators take one orthonormal frame into another they correspond to a change of basis For example if we have some orthonormal basis gr1v gt then for any unitary operator 3 we may construct another basis gt which is also orthonormal gt23mgt then I I 39 lt Ejg i gt lt EjgsT336139 gt 315 4 The length of any vector 311 gt is then invariant under such a change of basis although its components will change Unitary changes of basis are just analogous to using a rotated frame of reference in a discussion of two or three dimensional vectors Eigenvalues and Eigenvectors Consider again vectors in ordinary three dimensional space And consider the rotation operator most vectors will change under rotation the exception are vectors which lie along the axis of rotation They don t change at all Also vectors lying in the plane at 90 degrees to the axis just flip sign if the angle of rotation is 180 degrees In a complex vector space such used by QM every linear transformation has special vectors such as these they are called eigenveetars The transform into multiples of themselves under the operatortransformation The multiplying constant is called the eigenvalue T Q gt Na gt Hermitian operators are special because 0 They have real eigenvalues 0 Their eigenvectors are orthogonal and can be made orthonormal 0 They span the space that is any vector can be expanded a linear combination of the eigenvectors they can hence be used a basis set The last statement is strictly only always true for nite dimensional vector spaces The rst of these is easily proved lt a T a gt A lt 04a gt because of hermiticity we may rewrite this lt Toeoz gt N lt 04a gt we have used the fact that the Hermitian adjoint of a scalar is just its complex conjugate Hence A N QED To prove the second statement suppose T a gt Na gt 0 and Ti8 gt 148 gt Thus lt 04T8 gt u lt 048 gt Using hermiticity we see that this can be rewritten lt T048 gt u lt 048 gt But the LHS is just lt A048 gt and since A is real and u A we see lt 048 gt 0 Generalized Statistical Interpretation Suppose now that we have a quantum system described by a state vector 31 gt evolving according to eqn 1 It is a postulate of QM that every observable will be represented in the theory by an hermitian operator Suppose we choose to make a measurement of some physical observable corresponding to an hermitian operator Q The possible results of that measurement are the eigenvalues qi of Q and the probability of measuring qi is simply lt Pgt After the measurement the state collapses to the state 3 gt and will then continue to evolve according to eqn 1 once more
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