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Crash Course: QM Math Category: School Work (/category/School-Work) | Nov 25, 2017
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Description Just some math necessary for QM understanding.
Share Tags Eigenvalues And Eigenvectors (/tag/eigenvalues-and-eigenvectors) Operator (Mathematics) (/tag/operator-mathematics) Matrix (Mathematics) (/tag/matrix-mathematics) Linear Algebra (/tag/linear-algebra)
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Crash Course: The Math of Quantum Mechanics March 8 , 2013 Q uantum mechanics uses a wide range of mathematics including See also: http://simple.wikipedia.org/wiki/Quantum_mechanics complex numbers, linear algebra, calculus, and differential equa-tions. In this handout, we give a crash course on some of the Nucleic Acids: A crash course basicmathematics used. Complex Numbers Complex numbers 1 give us a way of saying that the (/nucleic-acids-a-crash-course) square of a num- 1 For another simple explanation, seealso: http://simple.wikipedia.org/wiki/Complex_numbers ber can be negative. To that end, we have to introduce the imaginarynumber i such that i = √ − 1.( 1 )A complex number then has the form z = a Crash Course on Machine + ib where a and b are the The number z can be treated just as anynumber with a variable i , you Learning Part VI (/crash-coursejust haveto remember that i 2 = − 1. usual real numbers we use every day.Every complex number on-machine-learning-part-vi) also has a complex conjugate . This numberis made by going through a complex number and replacing i with − i and is represented in physics by z * = a − ib . One important propertyof this Social Media Crash Course in conjugation is z * z = ( a − ib )( a + ib ) = a 2 + iab − iab − i 2 b 2 = a 2 + b 2 .( 2 )Since a and b Personal Branding (/socialare the usual real numbers, z * z is not only real, butpositive. Exponentiation A very important media-crash-course-in-personaloperation for complex numbers and many physicalapplications is the exponential function 2 . We branding) define it with an infinite 2 http://simple.wikipedia.org/wiki/Exponential_function sum e x = ∞ ∑ n = 0 Figurative Language Crash x n n ! = 1 + x 1 + x 2 2 · 1 + x 3 3 · 2 · 1 + ··· .( 3 )It has the important property that if you take its Course!!! (/figurative-languagederivative you get back the same function 3 3 We will see this a lot in the form ddt e − i t = − i e − crash-course) i t or ddx e ipx = ipe ipx . ddxe x = e x .( 4 )Now, this function has the particular property that if instead of x ,you use an imaginary number, you obtain This property gives the famous e i = − 1. C++ crash course (/c-crashe i = cos + i sin .( 5 )And in fact, any complex number can be represented by z = r e i where r course) and are both real numbers. We can see from this that z * z = Remember to change i to − i , so z * = re − i , and also remember that e x e y = e x + y . r 2 , so if we look at Eq.( 2 ), we can see that r = √ a 2 + b 2 . crash course : the math of quantum mechanics 2 Linear algebra Linear Crash Course Review (/crashalgebra is at the heart of quantum mechanics. It deals withvectors and matrices in many course-review) dimensions. Vectors A column vector defined by | v is defined as just an ordered set of The |· is called a “ket” in quantummechanics numbers in a column: | v = z 1 z 2 ... z N .( 6 )The number N is called the dimension . We can define the correspond-ing row vector by v | , and it Crash Course Review (/crashlooks like The ·| is called a “bra” in quantummechanics. v | = z * 1 z * 2 ··· z * N .( 7 )Notice course-review-bvaaUoy) how in making the column vector into a row vector we tookthe complex conjugate of each entry. This is very important since nowwe will define the inner product of a two vectors | v and | w as v | w (just replace z with w to obtain the entries of | w ) where for the abovewe get v | w = z * 1 w 1 Crash course for Species + z * 2 w 2 + ··· + z * N w N .( 8 ) Matrices Now, what makes linear algebra linear is that we act Relationships and Interaction on vectors with When we write a | v , it means multiplyeach entry of | v by the number a matrices (/crash-course-for-speciesthat have the property that if M is a matrix, | v and | w arevectors and a , b are just numbers then relationships-and-interaction) Course Credit (/search/course+credit) M ( a | v + b | w ) = aM | v + bM | w .A matrix looks like 4 4 Strictly speaking, this is a square CVS Concurrent Versioning ma-trix ; in general, the rows and columnsneed not be the same, but most of ouruse will be with System CRASH COURSE (/cvsMassive Open Online Course (/search/massive+open+online+course) square matrices. M = m 11 m 12 ··· m 1 N m 21 m 22 ··· m 2 N ............ m N 1 m N 2 ··· m concurrent-versioning-systemNN .( 9 )Matrices can be multipled, but the procedure for this is similar to thatof the inner crash-course) Golf Course (/search/golf+course) Course Evaluation (/search/course+evaluation) product and not how one might guess to multiply matri-ces. To develop this, consider two ways of ELEKS DevTalks #4: Amazon Wall Street Crash Of 1929 (/search/wall+street+crash+of+1929) thinking of a matrix: As arow vector of column vectors or as a column vector of row vectors: crash Web Services Crash Course course : the math of quantum mechanics 3 Here we have defined for example | m 1 = m 11 m (/eleks-devtalks-4-amazon-webStock Market Crash (/search/stock+market+crash) 21 ... m N 1 and ˜ m 1 | = m 11 m 12 ··· m 1 N M = | m 1 | m 2 ··· | m N = ˜ m 1 | ˜ services-crash-course) m 2 | ... ˜ m N | .( 10 )And like this we can define matrix multiplication MN = ˜ m 1 | ˜ m Crash Course in PR 101 (no 2 | ... ˜ m N | | n 1 | n 2 ··· | n N = ˜ m 1 | n 1 ˜ m 1 | n 2 ··· ˜ m 1 | n N ˜ m 2 | helmet needed!) (/crash-coursen 1 ˜ m 2 | n 2 ··· ˜ m 2 | n N ............ ˜ m N | n 1 ˜ m N | n 2 ··· ˜ m N | n N .( 11 in-pr-101-no-helmet-needed) )For a real example of this, see 5 . 5 http://simple.wikipedia.org/wiki/Matrix_(mathematics) Multiplication by a vector is similar. In fact we can define it forcolumn and row vectors: M | v = A Crash Course in Finding ˜ m 1 | ˜ m 2 | ... ˜ m N | | v = ˜ m 1 | v ˜ m 2 | v ... ˜ m N | v ,( 12 )and v | Vacant houses and Off Market M = v | | m 1 | m 2 ··· | m N ( 13 ) = v | m 1 v | m 2 ··· v | m N Now, we can also Property! (/a-crash-course-inconstruct the hermitian conjugate of a matrix, M by just writing M † = m * 11 m * 21 ··· m * N finding-vacant-houses-and-off1 m * 12 m * 22 ··· m * 2 N ............ m * 1 N m * 2 N ··· m * NN .( 14 )Notice how every Company Crash Course market-property) entry is the complex conjugate and flipped acrossthe diagonal 6 . A matrix is hermitian if M = M † . (/company-crash-course) Note that a hermitian 6 This is also sometimes called theconjugate transpose. matrix doesn’t necessarily have all real entries, just that the entriesabove the diagonal are conjugate to those below the diagonal, asdefined by m * ij = m ji .There is also a special matrix called the identity View more... matrix which has 1 crash course : the math of quantum mechanics 4 on the diagonal and zero off (/search/Crash+Course+QM+Math) the diagonal: I = 1 0 ··· 00 1 ··· 0............0 0 ··· 1 ,( 15 )this has the property that MI = M and IM = M for any matrix M (Check this as an exercise). Eigenvalues and Eigenvectors Matrices Referat Eritroderma (/referat-eritroderma) STUDIS - Makalah (Kel.10) Kerajaan-kerajaan Islam di Indonesia.docx can have special vectors called eigenvectors which have theproperty that if | E is an eigenvector of M , M | E = E | E ,( 16 )we sometimes identify eigenvectors by their eigenvalues , calling | E , Note that if | E is an eigenvector, sois a | E . This ambiguiuty oftentimeslets us normalize the vectors by usingthe eigenvector with E | E = 1. Thiscondition is also related to probabilitiesin quantum mechanics. the vector with eigenvalue E . If M is hermitian, as it usually is in ourapplications, then the eigenvalues E are real numbers (not imaginaryor complex).And if M is
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