PERSAMAAN SCHRODINGER PDF

View My Stats. Metode Nikivarof Uvarov merupakan metode penyelesaian persamaan diferensial orde dua dengan mengubah persamaan diferensial orde dua yang umum persamaan Schrodinger menjadi persamaan diferensial tipe hipergeometrik melalui substitusi variabel yang sesuai untuk memperoleh eigen value dan fungsi gelombang bagian sudut. Pada penelitian ini bertujuan untuk mengetahui bagaimana fungsi gelombang bagian sudut persamaan schrodinger D-dimensi untuk potensial Poschl-Teller Hiperbolik Terdeformasi q plus Rosen Morse Trigonometri Terdeformasi q menggunakan metode Nikiforov-Uvarov NU. Nikivarof Uvarov is a method to solve second order differential equations by changing general second order differential equation to hyper-geometric differential equation type through substituting relevant variable to obtain eigenvalues and the angle of wave function.

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No notes for slide. Persamaan schroedinger bebas waktu 1. Persamaan SchroedingerPostulat-postulat dasar Mekanika KuantumPostulat I: Setiap sistem fisis dinyatakan dengan fungsi gelombang atau fungsikeadaan, , yang secara implisit memuat informasi lengkapmengenai observabel-observabel yang dapat diketahui pada sistemtersebut.

Postulat II: Setiap observabel dinyatakan atau diwakili oleh suatu operatorlinear hermitan. OperatorOperator adalah suatu instruksi matematis yang bila dikenakan ataudioperasikan pada suatu fungsi maka akan mengubah fungsi tersebut menjadifungsi lain. Beberapa Operator Observabel No. Observabel Operator 1. Posisi 2. Momentum Linier 3. Momentum Sudut 4. Energi Kinetik 2.

Persamaan Schroedinger 5. Energi Potensial 6. Energi TotalPersamaan schroedinger bebas waktuJika fungsi potensial tidak bergantung waktu, bagaimanakah bentuk persamaanSchroedinger untuk kasus dengan potensial bebas waktu V x? Untuk kasus seperti itu persamaan gelombang SchroedingerBila dilakukan separasi variable pemisahan peubah dalam solusi persamaan di atassehingga lalu substitusikan dalam persamaan Schroedinger bebaswaktu menghasilkan :dan dapat ditulis pula kedalam bentuk :Dari persamaan di atas jelas terlihat bahwa ruas kiri dari persamaan tersebut hanyamengandung variable x, dan ruas tengah hanya mengandung variable t.

Sedangkanpersamaan itu berlaku untuk semua harga x maupun t. Hal ini hanya berlaku jika ruas kiridan ruas tengah selalu bernilai tetap, misalkan sama dengan G.

Dengan demikian dapat diperoleh dua persamaan berikut : 3. Berikut penjelasannya :Perhatikan persamaan dan lalubandingkandengan persamaan maka didapat ungkapan :sehingga otomatis nilai G sama besarnya dengan energi total partikel E.

Dengan demikianuntuk kasus dengan fungsi potensial tidak bergantung waktu, diperoleh persamaanSchroedinger bebas waktu PSBW :dengan fungsi gelombang total:persamaan , yang dapat ditulis sebagai dinamakan persamaan harga eigen, dan harga tetap E yangmerupakan solusi yang dikenal sebagai nama persamaan karakteristik, suatu topik pentingdalam pembelajaran tentang persamaan diferensial.

Sumur Potensial Persegi Tak Terhingga 4. Persamaan SchroedingerAndaikanlah suatu elektron dalam pengaruh potensial berbentuk sumur tak terhinggaberdimensi-1 seperti berikutElektron terperangkap dalam daerah — , dan sama sekali tak dapat ke luar daerahitu.

Dengan perkata lain peluang elektron berada di dan di sama dengan nol. Persamaan SchroedingerFungsi-fungsi ini membentuk set ortonormal; artinya:Selanjutnya, diperoleh harga eigen energi:Energi ini berharga diskrit tidak kontinu, tapi bertingkat-tingkat ditandai oleh bilangankuantum n. Theinfinite square well potential is illustrated in Fig. Inorder to normalize the wave functions, the constant A must be determined.

One obtains theeigenstate energies in the infinite square well according toThe spacing between two adjacent energy levels, that is En — En-1, is proportional to n. Thus,the energetic spacing between states increases with energy. The energy levels areschematicallyshown in Fig.

The probabilitydensities of the three lowest states are shown in Fig. The eigenstate energies are, asalready mentioned, expectation values of the total energy of the respective state. It istherefore interesting to know if the eigenstate energies are purely kinetic, purely potential,or a mixture of both.

The expectation value of the kinetic energy of the ground state iscalculated according to the 5th Postulate: 8. Evaluation of kinetic energies of allother states yieldsThe kinetic energy coincides with the total energy given in Eq. Thus, the energy of aparticle in an infinite square well is purely kinetic.

The particle has no potential energy. We next turn to a second, more intuitive method to obtain the wave functions of the infinitepotential well.

This second method is based on the de Broglie wave concept. Recall that thede. Broglie wave is defined for a constant momentum p, that is for a particle in a constantpotential. The energy of the wave is purely kinetic. In order to find solutions of the infinite squarewell, wematch the de Broglie wavelength to the width of the quantum well according to theconditionIn this equation, multiples of half of the de Broglie wavelength are matched to the width ofthe quantum well.

Expressing the kinetic energy in terms of the de Broglie wavelength, thatisand inserting Eq. Persamaan SchroedingerThis equation is identical to Eq. The de Broglie wave concept yields the correct solution of theinfinite potential well,because i the particle is confined to the constant potential of the well region, ii theenergy ofparticle is purely kinetic, and iii the wave function is sinusoidal.

The infinite square shaped quantum well is the simplest of all potential wells. The wavefunctions eigenfunctions and energies eigenvalues in an infinite square well arerelativelysimple.

There is a large number of potential wells with other shapes, for example thesquare wellwith finite barriers, parabolic well, triangular well, or V-shaped well. The exact solutions ofthese wells are more complicated. Several methods have been developed to calculateapproximate solutions for arbitrary shaped potential wells. These methods will bediscussed in the Chapter on quantum wells in this book. The asymmetric and symmetric finite square-shaped quantum wellIn contrast to the infinite square well, the finite square well has barriers of finite height.

Thepotential of a finite square well is shown in Fig. The two barriers of the well have adifferent height and therefore, the structure is denoted asymmetric square well.

Persamaan SchroedingerAssume that a particle with energy E is in a constant potential U. Next, the solutions of an asymmetric and symmetric square well will be calculated. The Persamaan Schroedingerpotential energy of the well is piecewise constant, as shown in Fig. From this condition, one obtains ……………………………………….. For the finitesymmetric square well, which is of great practical relevance, the eigenvalue equation isgiven by : ………………………………………….

If K is expressed as a function of k see Eq. Solving the eigenvalue equation yields the eigenvalues of kand, by using Eqs. The allowed energies are also called the eigenstate energies of the potential.

Inspection of Eq. Non-trivial solutions of theeigenvalue equation can be obtained by a graphical method. Figure 7. The dashed curve represents Persamaan Schroedingerthe right-hand side of the eigenvalue equation. The intersections of the dashed curve withthe periodic tangent function are the solutions of the eigenvalue equation.

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Schrodinger Equation

The Schrodinger equation plays the role of Newton's laws and conservation of energy in classical mechanics - i. It is a wave equation in terms of the wavefunction which predicts analytically and precisely the probability of events or outcome. The detailed outcome is not strictly determined, but given a large number of events, the Schrodinger equation will predict the distribution of results. The kinetic and potential energies are transformed into the Hamiltonian which acts upon the wavefunction to generate the evolution of the wavefunction in time and space. The Schrodinger equation gives the quantized energies of the system and gives the form of the wavefunction so that other properties may be calculated.

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Fungsi gelombang

Fungsi gelombang dalam fisika kuantum adalah suatu persamaan matematis yang menggambarkan keadaan kuantum dari suatu sistem kuantum terisolasi. Fungsi gelombang merupakan suatu amplitudo probabilitas bernilai-kompleks , dan kebolehjadian untuk hasil yang mungkin dari pengukuran yang dibuat oleh sistem dapat diturunkan darinya. Secara umum, fungsi gelombang suatu sistem dapat dinyatakan dalam berbagai peubah, seperti dalam momentum , posisi, energi , dan sebagainya. Fungsi gelombang dapat pula berupa fungsi waktu , dan dapat pula dinyatakan sebagai fungsi tak-gayut waktu. Menurut prinsip superposisi mekanika kuantum, fungsi gelombang dapat dijumlahkan dan dikali dengan bilangan kompleks untuk menghasilkan fungsi gelombang baru dan suatu ruang Hilbert. Hasil kali antara dua fungsi gelombang merupakan ukuran tumpang-tindih antara keadaan fisika terkait, dan digunakan sebagai dasar interpretasi kebolehjadian pada mekanika kuantum, hukum Born , yang mengaitkan kebolehjadian transisi pada hasil kali tersebut.

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